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+\section{Primitive of Cube of Hyperbolic Tangent of a x/Proof 2}
+Tags: Primitives involving Hyperbolic Tangent Function
+
+\begin{theorem}
+:$\displaystyle \int \tanh^3 a x \ \mathrm d x = \frac {\ln \left\vert{\cosh a x}\right\vert} a - \frac {\tanh^2 a x} {2 a} + C$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \int \tanh^3 a x \ \mathrm d x
+      | r = - \frac {\tanh^2 a x} {2 a} + \int \tanh a x \ \mathrm d x
+      | c = [[Primitive of Power of Hyperbolic Tangent of a x|Primitive of Power of $\tanh^n a x$]] with $n = 3$
+}}
+{{eqn | r = \frac {\ln \left\vert{\cosh a x}\right\vert} a - \frac {\tanh^2 a x} {2 a} + C
+      | c = [[Primitive of Hyperbolic Tangent of a x|Primitive of $\tanh a x$]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of Cube of Hyperbolic Cotangent of a x}
+Tags: Primitive of Cube of Hyperbolic Cotangent of a x
+
+\begin{theorem}
+:$\ds \int \coth^3 a x \rd x = \frac {\ln \size {\sinh a x} } a - \frac {\coth^2 a x} {2 a} + C$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \int \coth^3 a x \ \mathrm d x
+      | r = \int \coth a x \coth^2 a x \ \mathrm d x
+      | c = 
+}}
+{{eqn | r = \int \coth a x \left({1 + \operatorname{csch}^2 a x}\right) \ \mathrm d x
+      | c = [[Difference of Squares of Hyperbolic Cotangent and Cosecant]]
+}}
+{{eqn | r = \int \coth a x \ \mathrm d x + \int \coth a x \operatorname{csch}^2 a x \ \mathrm d x
+      | c = [[Linear Combination of Integrals]]
+}}
+{{eqn | r = \frac {\ln \left\vert{\sinh a x}\right\vert} a + \int \coth a x \operatorname{csch}^2 a x \ \mathrm d x
+      | c = [[Primitive of Hyperbolic Tangent of a x|Primitive of $\tanh a x$]]
+}}
+{{eqn | r = \frac {\ln \left\vert{\sinh a x}\right\vert} a - \frac {\coth^2 a x} {2 a} + C
+      | c = [[Primitive of Power of Hyperbolic Cotangent of a x by Square of Hyperbolic Cosecant of a x|Primitive of $\coth^n a x \operatorname{csch}^2 a x$]]: $n = 1$
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \int \coth^3 a x \ \mathrm d x
+      | r = - \frac {\coth^2 a x} {2 a} + \int \coth a x \ \mathrm d x
+      | c = [[Primitive of Power of Hyperbolic Cotangent of a x|Primitive of Power of $\coth^n a x$]] with $n = 3$
+}}
+{{eqn | r = \frac {\ln \left\vert{\sinh a x}\right\vert} a - \frac {\coth^2 a x} {2 a} + C
+      | c = [[Primitive of Hyperbolic Cotangent of a x|Primitive of $\coth a x$]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of Cube of Hyperbolic Cotangent of a x/Proof 1}
+Tags: Primitive of Cube of Hyperbolic Cotangent of a x
+
+\begin{theorem}
+:$\displaystyle \int \coth^3 a x \ \mathrm d x = \frac {\ln \left\vert{\sinh a x}\right\vert} a - \frac {\coth^2 a x} {2 a} + C$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \int \coth^3 a x \ \mathrm d x
+      | r = \int \coth a x \coth^2 a x \ \mathrm d x
+      | c = 
+}}
+{{eqn | r = \int \coth a x \left({1 + \operatorname{csch}^2 a x}\right) \ \mathrm d x
+      | c = [[Difference of Squares of Hyperbolic Cotangent and Cosecant]]
+}}
+{{eqn | r = \int \coth a x \ \mathrm d x + \int \coth a x \operatorname{csch}^2 a x \ \mathrm d x
+      | c = [[Linear Combination of Integrals]]
+}}
+{{eqn | r = \frac {\ln \left\vert{\sinh a x}\right\vert} a + \int \coth a x \operatorname{csch}^2 a x \ \mathrm d x
+      | c = [[Primitive of Hyperbolic Tangent of a x|Primitive of $\tanh a x$]]
+}}
+{{eqn | r = \frac {\ln \left\vert{\sinh a x}\right\vert} a - \frac {\coth^2 a x} {2 a} + C
+      | c = [[Primitive of Power of Hyperbolic Cotangent of a x by Square of Hyperbolic Cosecant of a x|Primitive of $\coth^n a x \operatorname{csch}^2 a x$]]: $n = 1$
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of Cube of Hyperbolic Cotangent of a x/Proof 2}
+Tags: Primitive of Cube of Hyperbolic Cotangent of a x
+
+\begin{theorem}
+:$\displaystyle \int \coth^3 a x \ \mathrm d x = \frac {\ln \left\vert{\sinh a x}\right\vert} a - \frac {\coth^2 a x} {2 a} + C$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \int \coth^3 a x \ \mathrm d x
+      | r = - \frac {\coth^2 a x} {2 a} + \int \coth a x \ \mathrm d x
+      | c = [[Primitive of Power of Hyperbolic Cotangent of a x|Primitive of Power of $\coth^n a x$]] with $n = 3$
+}}
+{{eqn | r = \frac {\ln \left\vert{\sinh a x}\right\vert} a - \frac {\coth^2 a x} {2 a} + C
+      | c = [[Primitive of Hyperbolic Cotangent of a x|Primitive of $\coth a x$]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of x by Hyperbolic Tangent of a x}
+Tags: Primitives involving Hyperbolic Tangent Function
+
+\begin{theorem}
+:$\displaystyle \int x \tanh a x \rd x = \frac 1 {a^2} \paren {\frac {\paren {a x}^3} 3 - \frac {\paren {a x}^5} {15} + \frac {2 \paren {a x}^7} {105} + \cdots + \frac { 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn 	| l = \int x \tanh a x \rd x 
+	| r = \frac 1 {a^2} \int \theta \tanh \theta \rd \theta 
+	| c = [[Integration by Substitution|Substitution of $a x \to \theta$]]
+}}
+{{eqn 	| r = \frac 1 {a^2} \int \theta \sum_{n \mathop = 0}^\infty \frac{2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \, \theta^{2 n - 1} } {\paren {2 n}!} \rd \theta
+	| c = [[Power Series Expansion for Hyperbolic Tangent Function]]
+}}
+{{eqn 	| r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {2^{2 n} \paren {2^{2 n} - 1} B_{2 n} } {\paren {2 n}!} \int \theta^{2 n} \rd \theta
+	| c = [[Fubini's Theorem]]
+}}
+{{eqn 	| r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!}  + C
+	| c = [[Integration by Substitution|Substituting back $\theta \to ax$]]
+}} 
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of x by Hyperbolic Cotangent of a x}
+Tags: Primitives involving Hyperbolic Cotangent Function
+
+\begin{theorem}
+:$\displaystyle \int x \coth a x \rd x = \frac 1 {a^2} \paren {a x + \frac {\paren {a x}^3} 9 - \frac {\paren {a x}^5} {225} + \cdots + \frac {2^{2 n} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn 	| l = \int x \coth a x \rd x 
+	| r = \frac 1 {a^2} \int \theta \coth \theta \rd \theta 
+	| c = [[Integration by Substitution|Substitution of $a x \to \theta$]]
+}}
+{{eqn 	| r = \frac 1 {a^2} \int \theta \sum_{n \mathop = 0}^\infty \frac{2^{2 n} B_{2 n} \, \theta^{2 n - 1} } {\paren {2 n}!} \rd \theta
+	| c = [[Power Series Expansion for Hyperbolic Cotangent Function]]
+}}
+{{eqn 	| r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {2^{2 n} B_{2 n} } {\paren {2 n}!} \int \theta^{2 n} \rd \theta
+	| c = [[Fubini's Theorem]]
+}}
+{{eqn 	| r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {2^{2 n} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!}  + C
+	| c = [[Integration by Substitution|Substituting back $\theta \to ax$]]
+}} 
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of Square of Hyperbolic Cosecant of a x over Hyperbolic Cotangent of a x}
+Tags: Primitives involving Hyperbolic Cotangent Function, Primitives involving Hyperbolic Cosecant Function
+
+\begin{theorem}
+:$\displaystyle \int \frac {\csch^2 a x \rd x} {\coth a x} = \frac {-1} a \ln \size {\coth a x} + C$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \frac \d {\d x} \coth a x
+      | r = -a \csch^2 a x
+      | c = [[Derivative of Hyperbolic Cotangent of a x|Derivative of $\coth a x$]]
+}}
+{{eqn | ll= \leadsto
+      | l = \int \frac {-a \csch^2 a x \rd x} {\coth a x}
+      | r = \ln \size {\coth a x} + C
+      | c = [[Primitive of Function under its Derivative]]
+}}
+{{eqn | ll= \leadsto
+      | l = \int \frac {\csch^2 a x \rd x} {\coth a x}
+      | r = \frac {-1} a \ln \size {\coth a x} + C
+      | c = [[Primitive of Constant Multiple of Function]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of Cube of Hyperbolic Secant of a x}
+Tags: Primitives involving Hyperbolic Secant Function
+
+\begin{theorem}
+:$\displaystyle \int \sech^3 a x \rd x = \frac {\sech a x \tanh a x} {2 a} + \frac 1 {2 a} \map \arctan {\sinh a x} + C$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \int \sech^3 x \rd x
+      | r = \frac {\sech a x \tanh a x} {2 a} + \frac 1 2 \int \sech a x \rd x
+      | c = [[Primitive of Power of Hyperbolic Secant of a x|Primitive of $\sech^n a x$]] where $n = 3$
+}}
+{{eqn | r = \frac {\sech a x \tanh a x} {2 a} + \frac 1 {2 a} \map \arctan {\sinh a x} + C
+      | c = [[Primitive of Hyperbolic Secant of a x/Arctangent of Hyperbolic Sine Form|Primitive of $\sech a x$]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of Cube of Hyperbolic Cosecant of a x}
+Tags: Primitives involving Hyperbolic Cosecant Function
+
+\begin{theorem}
+:$\ds \int \csch^3 a x \rd x = \frac {-\csch a x \coth a x} {2 a} - \frac 1 {2 a} \ln \size {\tanh a x} + C$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \int \csch^3 x \rd x
+      | r = \frac {\csch a x \coth a x} {2 a} - \frac 1 2 \int \csch a x \rd x
+      | c = [[Primitive of Power of Hyperbolic Cosecant of a x|Primitive of $\csch^n a x$]] where $n = 3$
+}}
+{{eqn | r = \frac {\csch a x \coth a x} {2 a} - \ln \size {\tanh a x} + C
+      | c = [[Primitive of Hyperbolic Cosecant of a x|Primitive of $\csch a x$]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of Hyperbolic Secant Function/Arctangent of Hyperbolic Sine Form}
+Tags: Primitive of Hyperbolic Secant Function
+
+\begin{theorem}
+:$\ds \int \sech x \rd x =  \map \arctan {\sinh x} + C$
+\end{theorem}
+
+\begin{proof}
+We have that:
+{{begin-eqn}}
+{{eqn | l = \int \sech x \rd x
+      | r = \int \frac {\d x} {\cosh x}
+      | c = {{Defof|Hyperbolic Secant|index = 2}}
+}}
+{{eqn | r = \int \frac {\cosh x \rd x} {\cosh^2 x}
+      | c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $\cosh x$
+}}
+{{eqn | r = \int \frac {\cosh x \rd x} {1 + \sinh^2 x}
+      | c = [[Difference of Squares of Hyperbolic Cosine and Sine]]
+}}
+{{end-eqn}}
+Let:
+{{begin-eqn}}
+{{eqn | l = u
+      | r = \sinh x
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = \frac {\d u} {\d x}
+      | r = \cosh x
+      | c = [[Derivative of Hyperbolic Sine]]
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = \int \sech x \rd x
+      | r = \int \frac {\d u} {u^2 + 1}
+      | c = [[Integration by Substitution]]
+}}
+{{eqn | r = \arctan u + C
+      | c = [[Primitive of Reciprocal of x squared plus a squared/Arctangent Form|Primitive of $\dfrac 1 {x^2 + a^2}$: Arctangent Form]]
+}}
+{{eqn | r = \map \arctan {\sinh x} + C
+      | c = substituting for $u$
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of Hyperbolic Secant of a x/Arcsine Form}
+Tags: Primitives involving Hyperbolic Secant Function
+
+\begin{theorem}
+:$\displaystyle \int \operatorname{sech} a x \ \mathrm d x = \frac {\arcsin \left({\tanh a x}\right)} a + C$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \int \operatorname{sech} x \ \mathrm d x
+      | r = \arcsin \left({\tanh x}\right)
+      | c = [[Primitive of Hyperbolic Secant Function/Arcsine Form|Primitive of $\operatorname{sech} x$: Arcsine form]]
+}}
+{{eqn | ll= \implies
+      | l = \int \operatorname{sech} a x \ \mathrm d x
+      | r = \frac 1 a \left({\arcsin \left({\tanh a x}\right) }\right) + C
+      | c = [[Primitive of Function of Constant Multiple]]
+}}
+{{eqn | r = \frac {\arcsin \left({\tanh a x}\right)} a + C
+      | c = simplifying
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Primitives involving Hyperbolic Secant Function]]
+286k72unv29baibz0j9l8nr95l5511e
+\end{proof}<|endoftext|>
+\section{Primitive of Hyperbolic Secant of a x/Arctangent of Exponential Form}
+Tags: Primitives involving Hyperbolic Secant Function
+
+\begin{theorem}
+:$\ds \int \sech a x \rd x = \frac {2 \map \arctan {e^{a x} } } a + C$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \int \sech x \rd x
+      | r = 2 \map \arctan {e^x} + C
+      | c = [[Primitive of Hyperbolic Secant Function/Arctangent of Exponential Form|Primitive of $\sech x$: Arctangent of Exponential Form]]
+}}
+{{eqn | ll= \leadsto
+      | l = \int \sech a x \rd x
+      | r = \frac 1 a \paren {2 \map \arctan {e^{a x} } } + C
+      | c = [[Primitive of Function of Constant Multiple]]
+}}
+{{eqn | r = \frac {2 \map \arctan {e^{a x} } } a + C
+      | c = simplifying
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of Hyperbolic Secant of a x/Arctangent of Hyperbolic Sine Form}
+Tags: Primitives involving Hyperbolic Secant Function
+
+\begin{theorem}
+:$\displaystyle \int \operatorname{sech} a x \ \mathrm d x = \frac {\arctan \left({\sinh a x}\right)} a + C$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \int \operatorname{sech} x \ \mathrm d x
+      | r = \arctan \left({\sinh x}\right)
+      | c = [[Primitive of Hyperbolic Secant Function/Arctangent of Hyperbolic Sine Form|Primitive of $\operatorname{sech} x$: Arctangent of Hyperbolic Sine form]]
+}}
+{{eqn | ll= \implies
+      | l = \int \operatorname{sech} a x \ \mathrm d x
+      | r = \frac 1 a \left({\arctan \left({\sinh a x}\right) }\right) + C
+      | c = [[Primitive of Function of Constant Multiple]]
+}}
+{{eqn | r = \frac {\arctan \left({\sinh a x}\right)} a + C
+      | c = simplifying
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Primitives involving Hyperbolic Secant Function]]
+b2eqzaz6izdhq0m44q1069zyforgeoq
+\end{proof}<|endoftext|>
+\section{Primitive of x by Hyperbolic Secant of a x}
+Tags: Primitives involving Hyperbolic Secant Function
+
+\begin{theorem}
+:$\displaystyle \int x \sech a x \rd x = \frac 1 {a^2} \paren {\frac {\paren {a x}^2} 2 - \frac {\paren {a x}^4} 8 + \frac {5 \paren {a x}^6} {144} - \cdots + \frac {E_{2 n} \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!} + \cdots} + C$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn 	| l = \int x \sech a x \rd x 
+	| r = \frac 1 {a^2} \int \theta \sech \theta \rd \theta 
+	| c = [[Integration by Substitution|Substitution of $a x \to \theta$]]
+}}
+{{eqn 	| r = \frac 1 {a^2} \int \theta \sum_{n \mathop = 0}^\infty \frac{E_{2 n} \, \theta^{2 n} } {\paren {2 n}!} \rd \theta
+	| c = [[Power Series Expansion for Hyperbolic Secant Function]]
+}}
+{{eqn 	| r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {E_{2 n} } {\paren {2 n}!} \int \theta^{2 n + 1} \rd \theta
+	| c = [[Fubini's Theorem]]
+}}
+{{eqn 	| r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {E_{2 n} \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!}  + C
+	| c = [[Integration by Substitution|Substituting back $\theta \to ax$]]
+}} 
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of x by Hyperbolic Cosecant of a x}
+Tags: Primitives involving Hyperbolic Cosecant Function
+
+\begin{theorem}
+:$\displaystyle \int x \csch a x \rd x = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {2 \paren {1 - 2^{2 n - 1} } B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + C$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn 	| l = \int x \csch a x \rd x 
+	| r = \frac 1 {a^2} \int \theta \csch \theta \rd \theta 
+	| c = [[Integration by Substitution|Substitution of $a x \to \theta$]]
+}}
+{{eqn 	| r = \frac 1 {a^2} \int \theta \sum_{n \mathop = 0}^\infty \frac {2 \paren {1 - 2^{2 n - 1} } B_{2 n} \, \theta^{2 n - 1} } {\paren {2 n}!} \rd \theta
+	| c = [[Power Series Expansion for Hyperbolic Cosecant Function]]
+}}
+{{eqn 	| r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {2 \paren {1 - 2^{2 n - 1} } B_{2 n} } {\paren {2 n}!} \int \theta^{2 n} \rd \theta
+	| c = [[Fubini's Theorem]]
+}}
+{{eqn 	| r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {2 \paren {1 - 2^{2 n - 1} } B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!}  + C
+	| c = [[Integration by Substitution|Substituting back $\theta \to ax$]]
+}} 
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Derivative of Inverse Hyperbolic Sine of x over a}
+Tags: Derivative of Inverse Hyperbolic Sine
+
+\begin{theorem}
+:$\dfrac {\map \d {\map {\sinh^{-1} } {\frac x a} } } {\d x} = \dfrac 1 {\sqrt {x^2 + a^2}}$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \frac {\map \d {\map {\sinh^{-1} } {\frac x a} } } {\d x}
+      | r = \frac 1 a \frac 1 {\sqrt {\paren {\frac x a}^2} + 1}
+      | c = [[Derivative of Inverse Hyperbolic Sine|Derivative of $\sinh^{-1}$]] and [[Derivative of Function of Constant Multiple]]	
+}}
+{{eqn | r = \frac 1 a \frac 1 {\sqrt {\frac {x^2 + a^2} {a^2} } }
+      | c = 
+}}
+{{eqn | r = \frac 1 a \frac a {\sqrt {x^2 + a^2} }
+      | c = 
+}}
+{{eqn | r = \frac 1 {\sqrt {x^2 + a^2} }
+      | c = 
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of Inverse Hyperbolic Sine of x over a}
+Tags: Primitives involving Inverse Hyperbolic Sine Function
+
+\begin{theorem}
+:$\displaystyle \int \sinh^{-1} \frac x a \rd x = x \sinh^{-1} \frac x a - \sqrt {x^2 + a^2} + C$
+\end{theorem}
+
+\begin{proof}
+With a view to expressing the [[Definition:Primitive|primitive]] in the form:
+:$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
+let:
+{{begin-eqn}}
+{{eqn | l = u
+      | r = \sinh^{-1} \frac x a
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = \frac {\d u} {\d x}
+      | r = \frac 1 {\sqrt {x^2 + a^2} }
+      | c = [[Derivative of Inverse Hyperbolic Sine of x over a|Derivative of $\sinh^{-1} \dfrac x a$]]
+}}
+{{end-eqn}}
+and let:
+{{begin-eqn}}
+{{eqn | l = \frac {\d v} {\d x}
+      | r = 1
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = v
+      | r = x
+      | c = [[Primitive of Constant]]
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = \int \sinh^{-1} \frac x a \rd x
+      | r = x \sinh^{-1} \frac x a - \int x \paren {\frac 1 {\sqrt {x^2 + a^2} } } \rd x + C
+      | c = [[Integration by Parts]]
+}}
+{{eqn | r = x \sinh^{-1} \frac x a - \int \frac {x \rd x} {\sqrt {x^2 + a^2} } + C
+      | c = simplifying
+}}
+{{eqn | r = x \sinh^{-1} \frac x a - \sqrt {x^2 + a^2} + C
+      | c = [[Primitive of x over Root of x squared plus a squared|Primitive of $\dfrac x {\sqrt {x^2 + a^2} }$]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Derivative of Inverse Hyperbolic Cosine of x over a}
+Tags: Derivative of Inverse Hyperbolic Cosine
+
+\begin{theorem}
+:$\dfrac {\map \d {\map {\cosh^{-1} } {\frac x a} } } {\d x} = \dfrac 1 {\sqrt {x^2 - a^2} }$
+where $x > a$.
+\end{theorem}
+
+\begin{proof}
+Let $x > a$.
+Then $\dfrac x a > 1$ and so:
+{{begin-eqn}}
+{{eqn | l = \frac {\map \d {\map {\cosh^{-1} } {\frac x a} } } {\d x}
+      | r = \frac 1 a \frac 1 {\sqrt {\paren {\frac x a}^2} - 1}
+      | c = [[Derivative of Inverse Hyperbolic Cosine|Derivative of $\cosh^{-1}$]] and [[Derivative of Function of Constant Multiple]]	
+}}
+{{eqn | r = \frac 1 a \frac 1 {\sqrt {\frac {x^2 - a^2} {a^2} } }
+      | c = 
+}}
+{{eqn | r = \frac 1 a \frac a {\sqrt {x^2 - a^2} }
+      | c = 
+}}
+{{eqn | r = \frac 1 {\sqrt {x^2 - a^2} }
+      | c = 
+}}
+{{end-eqn}}
+$\cosh^{-1} \dfrac x a$ is not defined when $x \le a$.
+When $x = a$ we have that $\sqrt {x^2 - a^2} = 0$ and so $\dfrac 1 {\sqrt {x^2 - a^2} }$ is not defined.
+Hence the restriction on the [[Definition:Domain of Real Function|domain]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Derivative of Inverse Hyperbolic Tangent of x over a}
+Tags: Derivative of Inverse Hyperbolic Tangent
+
+\begin{theorem}
+:$\map {\dfrac \d {\d x} } {\map {\tanh^{-1} } {\dfrac x a} } = \dfrac a {a^2 - x^2}$
+where $-a < x < a$.
+\end{theorem}
+
+\begin{proof}
+Let $-a < x < a$.
+Then $-1 < \dfrac x a < 1$ and so:
+{{begin-eqn}}
+{{eqn | l = \map {\dfrac \d {\d x} } {\map {\tanh^{-1} } {\dfrac x a} }
+      | r = \frac 1 a \frac 1 {1 - \paren {\frac x a}^2}
+      | c = [[Derivative of Inverse Hyperbolic Tangent|Derivative of $\tanh^{-1}$]] and [[Derivative of Function of Constant Multiple]]	
+}}
+{{eqn | r = \frac 1 a \frac 1 {\frac {a^2 - x^2} {a^2} }
+      | c = 
+}}
+{{eqn | r = \frac 1 a \frac {a^2} {a^2 - x^2}
+      | c = 
+}}
+{{eqn | r = \frac a {a^2 - x^2}
+      | c = 
+}}
+{{end-eqn}}
+$\tanh^{-1} \dfrac x a$ is not defined when either $x \le -a$ or $x \ge a$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Derivative of Inverse Hyperbolic Cotangent of x over a}
+Tags: Derivative of Inverse Hyperbolic Cotangent
+
+\begin{theorem}
+:$\map {\dfrac \d {\d x} } {\map {\coth^{-1} } {\dfrac x a} } = \dfrac {-a} {x^2 - a^2}$
+where $x^2 > a^2$.
+\end{theorem}
+
+\begin{proof}
+Let $x^2 > a^2$.
+Then either $\dfrac x a < -1$ or $\dfrac x a > 1$ and so:
+{{begin-eqn}}
+{{eqn | l = \map {\dfrac \d {\d x} } {\map {\coth^{-1} } {\dfrac x a} }
+      | r = \frac 1 a \frac 1 {1 - \paren {\frac x a}^2}
+      | c = [[Derivative of Inverse Hyperbolic Cotangent|Derivative of $\coth^{-1}$]] and [[Derivative of Function of Constant Multiple]]	
+}}
+{{eqn | r = \frac 1 a \frac 1 {\frac {a^2 - x^2} {a^2} }
+      | c = 
+}}
+{{eqn | r = -\frac 1 a \frac {a^2} {x^2 - a^2}
+      | c = 
+}}
+{{eqn | r = \frac {-a} {x^2 - a^2}
+      | c = 
+}}
+{{end-eqn}}
+$\coth^{-1} \dfrac x a$ is not defined when $x^2 \le a^2$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Derivative of Inverse Hyperbolic Secant of x over a}
+Tags: Derivatives of Inverse Hyperbolic Functions, Inverse Hyperbolic Secant
+
+\begin{theorem}
+:$\dfrac {\mathrm d \left({\operatorname{sech}^{-1} \left({\frac x a}\right)}\right)} {\mathrm d x} = \dfrac {-a} {x \sqrt{a^2 - x^2} }$
+where $0 < x < a$.
+\end{theorem}
+
+\begin{proof}
+Let $0 < x < a$.
+Then $0 < \dfrac x a < 1$ and so:
+{{begin-eqn}}
+{{eqn | l = \frac {\mathrm d \left({\operatorname{sech}^{-1} \frac x a}\right)} {\mathrm d x}
+      | r = \frac 1 a \frac {-1} {\frac x a \sqrt{1 - \left({\frac x a}\right)^2} }
+      | c = [[Derivative of Inverse Hyperbolic Secant|Derivative of $\operatorname{sech}^{-1}$]] and [[Derivative of Function of Constant Multiple]]	
+}}
+{{eqn | r = \frac 1 a \frac {-a} {x \sqrt{\frac {a^2 - x^2} {a^2} } }
+      | c = 
+}}
+{{eqn | r = \frac {-a} {x \sqrt{a^2 - x^2} }
+      | c = 
+}}
+{{end-eqn}}
+$\operatorname{sech}^{-1} \dfrac x a$ is not defined when $x \le 0$ or $x \ge a$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Derivative of Inverse Hyperbolic Cosecant of x over a}
+Tags: Derivatives of Inverse Hyperbolic Functions, Inverse Hyperbolic Cosecant
+
+\begin{theorem}
+:$\dfrac {\map \d {\csch^{-1} } {\frac x a} } {\d x} = \dfrac {-a} {\size x \sqrt {a^2 + x^2} }$
+where $x \ne 0$.
+\end{theorem}
+
+\begin{proof}
+Let $0 < x < a$.
+Then $0 < \dfrac x a < 1$ and so:
+{{begin-eqn}}
+{{eqn | l = \frac {\map \d {\csch^{-1} } {\frac x a} } {\d x}
+      | r = \frac 1 a \dfrac {-1} {\size {\frac x a} \sqrt {1 + \paren {\frac x a}^2} }
+      | c = [[Derivative of Inverse Hyperbolic Cosecant|Derivative of $\csch^{-1}$]] and [[Derivative of Function of Constant Multiple]]	
+}}
+{{eqn | r = \frac 1 a \frac {-a} {\size x \sqrt {\frac {a^2 + x^2} {a^2} } }
+      | c = 
+}}
+{{eqn | r = \frac {-a} {\size x \sqrt{a^2 + x^2} }
+      | c = 
+}}
+{{end-eqn}}
+$\csch^{-1} \dfrac x a$ is not defined when $x = 0$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of Inverse Hyperbolic Cosine of x over a}
+Tags: Primitives involving Inverse Hyperbolic Cosine Function
+
+\begin{theorem}
+:$\displaystyle \int \cosh^{-1} \frac x a \ \mathrm d x = \begin{cases}
+x \cosh^{-1} \dfrac x a - \sqrt {x^2 - a^2} + C & : \cosh^{-1} \dfrac x a > 0 \\
+x \cosh^{-1} \dfrac x a + \sqrt {x^2 - a^2} + C & : \cosh^{-1} \dfrac x a < 0
+\end{cases}$
+\end{theorem}
+
+\begin{proof}
+With a view to expressing the [[Definition:Primitive|primitive]] in the form:
+:$\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$
+let:
+{{begin-eqn}}
+{{eqn | l = u
+      | r = \cosh^{-1} \frac x a
+      | c = 
+}}
+{{eqn | ll= \implies
+      | l = \frac {\mathrm d u} {\mathrm d x}
+      | r = \frac 1 {\sqrt {x^2 - a^2} }
+      | c = [[Derivative of Inverse Hyperbolic Sine of x over a|Derivative of $\cosh^{-1} \dfrac x a$]]
+}}
+{{end-eqn}}
+and let:
+{{begin-eqn}}
+{{eqn | l = \frac {\mathrm d v} {\mathrm d x}
+      | r = 1
+      | c = 
+}}
+{{eqn | ll= \implies
+      | l = v
+      | r = x
+      | c = [[Primitive of Constant]]
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = \int \cosh^{-1} \frac x a \ \mathrm d x
+      | r = x \cosh^{-1} \frac x a - \int x \left({\frac 1 {\sqrt {x^2 - a^2} } }\right) \ \mathrm d x + C
+      | c = [[Integration by Parts]]
+}}
+{{eqn | r = x \cosh^{-1} \frac x a - \int \frac {x \ \mathrm d x} {\sqrt {x^2 - a^2} } + C
+      | c = simplifying
+}}
+{{eqn | r = x \cosh^{-1} \frac x a - \sqrt {x^2 + a^2} + C
+      | c = [[Primitive of x over Root of x squared minus a squared|Primitive of $\dfrac x {\sqrt {x^2 - a^2} }$]]
+}}
+{{end-eqn}}
+{{finish|Analysis to be done for the negative case. The confusion here is that $\cosh^{-1} \frac x a$ is ''defined'' as being positive, like $\sqrt x$.}}
+\end{proof}<|endoftext|>
+\section{Primitive of Inverse Hyperbolic Tangent of x over a}
+Tags: Primitives involving Inverse Hyperbolic Tangent Function
+
+\begin{theorem}
+:$\displaystyle \int \tanh^{-1} \frac x a \rd x = x \tanh^{-1} \dfrac x a + \frac {a \ln \left({a^2 - x^2}\right)} 2 + C$
+\end{theorem}
+
+\begin{proof}
+With a view to expressing the [[Definition:Primitive|primitive]] in the form:
+:$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
+let:
+{{begin-eqn}}
+{{eqn | l = u
+      | r = \tanh^{-1} \frac x a
+      | c = 
+}}
+{{eqn | ll= \implies
+      | l = \frac {\d u} {\d x}
+      | r = \frac a {a^2 - x^2}
+      | c = [[Derivative of Inverse Hyperbolic Tangent of x over a|Derivative of $\tanh^{-1} \dfrac x a$]]
+}}
+{{end-eqn}}
+and let:
+{{begin-eqn}}
+{{eqn | l = \frac {\d v} {\d x}
+      | r = 1
+      | c = 
+}}
+{{eqn | ll= \implies
+      | l = v
+      | r = x
+      | c = [[Primitive of Constant]]
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = \int \tanh^{-1} \frac x a \rd x
+      | r = x \tanh^{-1} \frac x a - \int x \left({\frac a {a^2 - x^2} }\right) \rd x + C
+      | c = [[Integration by Parts]]
+}}
+{{eqn | r = x \tanh^{-1} \frac x a - a \int \frac {x \rd x} {a^2 - x^2} + C
+      | c = [[Primitive of Constant Multiple of Function]]
+}}
+{{eqn | r = x \tanh^{-1} \frac x a - a \left({- \frac 1 2 \ln \left({a^2 - x^2}\right)}\right) + C
+      | c = [[Primitive of x over a squared minus x squared|Primitive of $\dfrac x {a^2 - x^2}$]]
+}}
+{{eqn | r = x \tanh^{-1} \dfrac x a + \frac {a \ln \left({a^2 - x^2}\right)} 2 + C
+      | c = simplifying
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of Inverse Hyperbolic Cotangent of x over a}
+Tags: Primitives involving Inverse Hyperbolic Cotangent Function
+
+\begin{theorem}
+:$\displaystyle \int \coth^{-1} \frac x a \ \mathrm d x = x \coth^{-1} \dfrac x a + \frac {a \ln \left({x^2 - a^2}\right)} 2 + C$
+\end{theorem}
+
+\begin{proof}
+With a view to expressing the [[Definition:Primitive|primitive]] in the form:
+:$\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$
+let:
+{{begin-eqn}}
+{{eqn | l = u
+      | r = \coth^{-1} \frac x a
+      | c = 
+}}
+{{eqn | ll= \implies
+      | l = \frac {\mathrm d u} {\mathrm d x}
+      | r = \frac {-a} {x^2 - a^2}
+      | c = [[Derivative of Inverse Hyperbolic Cotangent of x over a|Derivative of $\coth^{-1} \dfrac x a$]]
+}}
+{{end-eqn}}
+and let:
+{{begin-eqn}}
+{{eqn | l = \frac {\mathrm d v} {\mathrm d x}
+      | r = 1
+      | c = 
+}}
+{{eqn | ll= \implies
+      | l = v
+      | r = x
+      | c = [[Primitive of Constant]]
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = \int \coth^{-1} \frac x a \ \mathrm d x
+      | r = x \tanh^{-1} \frac x a - \int x \left({\frac {-a} {x^2 - a^2} }\right) \ \mathrm d x + C
+      | c = [[Integration by Parts]]
+}}
+{{eqn | r = x \coth^{-1} \frac x a + a \int \frac {x \ \mathrm d x} {x^2 - a^2} + C
+      | c = [[Primitive of Constant Multiple of Function]]
+}}
+{{eqn | r = x \coth^{-1} \frac x a + a \left({\frac 1 2 \ln \left({x^2 - a^2}\right)}\right) + C
+      | c = [[Primitive of x over x squared minus a squared|Primitive of $\dfrac x {x^2 - a^2}$]]
+}}
+{{eqn | r = x \coth^{-1} \dfrac x a + \frac {a \ln \left({x^2 - a^2}\right)} 2 + C
+      | c = simplifying
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of Inverse Hyperbolic Secant of x over a}
+Tags: Primitives involving Inverse Hyperbolic Secant Function
+
+\begin{theorem}
+:$\displaystyle \int \sech^{-1} \frac x a \rd x = \begin{cases}
+x \sech^{-1} \dfrac x a + a \arcsin \dfrac x a + C & : \sech^{-1} \dfrac x a > 0 \\
+x \sech^{-1} \dfrac x a - a \arcsin \dfrac x a + C & : \sech^{-1} \dfrac x a < 0
+\end{cases}$
+\end{theorem}
+
+\begin{proof}
+With a view to expressing the [[Definition:Primitive|primitive]] in the form:
+:$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
+let:
+{{begin-eqn}}
+{{eqn | l = u
+      | r = \sech^{-1} \frac x a
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = \frac {\d u} {\d x}
+      | r = \frac {-a} {x \sqrt {a^2 - x^2} }
+      | c = [[Derivative of Inverse Hyperbolic Secant of x over a|Derivative of $\sech^{-1} \dfrac x a$]]
+}}
+{{end-eqn}}
+and let:
+{{begin-eqn}}
+{{eqn | l = \frac {\d v} {\d x}
+      | r = 1
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = v
+      | r = x
+      | c = [[Primitive of Constant]]
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = \int \sech^{-1} \frac x a \rd x
+      | r = x \sech^{-1} \frac x a - \int x \paren {\frac {-a} {x \sqrt {a^2 - x^2} } } \rd x + C
+      | c = [[Integration by Parts]]
+}}
+{{eqn | r = x \sech^{-1} \frac x a + a \int \frac {\d x} {\sqrt {a^2 - x^2} } + C
+      | c = [[Primitive of Constant Multiple of Function]]
+}}
+{{eqn | r = x \sech^{-1} \frac x a + \arcsin \frac x a + C
+      | c = [[Primitive of Reciprocal of Root of a squared minus x squared/Arcsine Form|Primitive of $\dfrac 1 {\sqrt {a^2 - x^2} }$]]
+}}
+{{end-eqn}}
+{{finish|Analysis to be done for the negative case. The confusion here is that $\sech^{-1} \frac x a$ is ''defined'' as being positive, like $\sqrt x$.}}
+\end{proof}<|endoftext|>
+\section{Primitive of Inverse Hyperbolic Cosecant of x over a}
+Tags: Primitives involving Inverse Hyperbolic Cosecant Function
+
+\begin{theorem}
+:$\displaystyle \int \operatorname{csch}^{-1} \frac x a \ \mathrm d x = \begin{cases}
+x \operatorname{csch}^{-1} \dfrac x a + a \sinh^{-1} \dfrac x a + C & : x > 0 \\
+x \operatorname{csch}^{-1} \dfrac x a - a \sinh^{-1} \dfrac x a + C & : x < 0
+\end{cases}$
+\end{theorem}
+
+\begin{proof}
+With a view to expressing the [[Definition:Primitive|primitive]] in the form:
+:$\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$
+let:
+{{begin-eqn}}
+{{eqn | l = u
+      | r = \operatorname{csch}^{-1} \frac x a
+      | c = 
+}}
+{{eqn | ll= \implies
+      | l = \frac {\mathrm d u} {\mathrm d x}
+      | r = \frac {-a} {\left\vert{x}\right\vert \sqrt{a^2 + x^2} }
+      | c = [[Derivative of Inverse Hyperbolic Cosecant of x over a|Derivative of $\operatorname{csch}^{-1} \dfrac x a$]]
+}}
+{{end-eqn}}
+and let:
+{{begin-eqn}}
+{{eqn | l = \frac {\mathrm d v} {\mathrm d x}
+      | r = 1
+      | c = 
+}}
+{{eqn | ll= \implies
+      | l = v
+      | r = x
+      | c = [[Primitive of Constant]]
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = \int \operatorname{csch}^{-1} \frac x a \ \mathrm d x
+      | r = x \operatorname{csch}^{-1} \frac x a - \int x \left({\frac {-a} {\left\vert{x}\right\vert \sqrt{a^2 + x^2} } }\right) \ \mathrm d x + C
+      | c = [[Integration by Parts]]
+}}
+{{eqn | r = x \operatorname{csch}^{-1} \frac x a + a \int \frac {x \ \mathrm d x} {\left\vert{x}\right\vert \sqrt{a^2 + x^2} } + C
+      | c = [[Primitive of Constant Multiple of Function]]
+}}
+{{eqn | r = x \operatorname{csch}^{-1} \frac x a \begin{cases} \displaystyle \mathop + a \int \frac {\mathrm d x} {\sqrt{a^2 + x^2} } + C & : x > 0 \\ \displaystyle \mathop - a \int \frac {\mathrm d x} {\sqrt{a^2 + x^2} } + C & : x < 0\end{cases}
+      | c = Definition of [[Definition:Absolute Value|Absolute Value]]
+}}
+{{eqn | r = \begin{cases} \displaystyle x \operatorname{csch}^{-1} \frac x a + a \sinh^{-1} \dfrac x a + C & : x > 0 \\ \displaystyle x \operatorname{csch}^{-1} \frac x a - a \sinh^{-1} \dfrac x a + C & : x < 0 \end{cases}
+      | c = [[Primitive of Reciprocal of Root of x squared plus a squared/Inverse Hyperbolic Sine Form|Primitive of $\dfrac 1 {\sqrt{a^2 + x^2} }$]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of x by Inverse Hyperbolic Sine of x over a}
+Tags: Primitives involving Inverse Hyperbolic Sine Function
+
+\begin{theorem}
+:$\displaystyle \int x \sinh^{-1} \frac x a \rd x = \paren {\frac {x^2} 2 + \frac {a^2} 4} \sinh^{-1} \frac x a - \frac {x \sqrt {x^2 + a^2} } 4 + C$
+\end{theorem}
+
+\begin{proof}
+With a view to expressing the [[Definition:Primitive|primitive]] in the form:
+:$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
+let:
+{{begin-eqn}}
+{{eqn | l = u
+      | r = \sinh^{-1} \frac x a
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = \frac {\d u} {\d x}
+      | r = \frac 1 {\sqrt {x^2 + a^2} }
+      | c = [[Derivative of Inverse Hyperbolic Sine of x over a|Derivative of $\sinh^{-1} \dfrac x a$]]
+}}
+{{end-eqn}}
+and let:
+{{begin-eqn}}
+{{eqn | l = \frac {\d v} {\d x}
+      | r = x
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = v
+      | r = \frac {x^2} 2
+      | c = [[Primitive of Power]]
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = \int x \sinh^{-1} \frac x a \rd x
+      | r = \frac {x^2} 2 \sinh^{-1} \frac x a - \int \frac {x^2} 2 \paren {\frac 1 {\sqrt {x^2 + a^2} } } \rd x + C
+      | c = [[Integration by Parts]]
+}}
+{{eqn | r = \frac {x^2} 2 \sinh^{-1} \frac x a - \frac 1 2 \int \frac {x^2 \rd x} {\sqrt {x^2 + a^2} } + C
+      | c = simplifying
+}}
+{{eqn | r = \frac {x^2} 2 \sinh^{-1} \frac x a - \frac 1 2 \paren {\frac {x \sqrt {x^2 + a^2} } 2 - \frac {a^2} 2 \sinh^{-1} \frac x a} + C
+      | c = [[Primitive of x squared over Root of x squared plus a squared/Inverse Hyperbolic Sine Form|Primitive of $\dfrac {x^2} {\sqrt {x^2 + a^2} }$]]
+}}
+{{eqn | r = \paren {\frac {x^2} 2 + \frac {a^2} 4} \sinh^{-1} \frac x a - \frac {x \sqrt {x^2 + a^2} } 4 + C
+      | c = simplifying
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Inverse Hyperbolic Cosine of x over a in Logarithm Form}
+Tags: Inverse Hyperbolic Cosine
+
+\begin{theorem}
+:$\cosh^{-1} \dfrac x a = \map \ln {x + \sqrt {x^2 - a^2} } - \ln a$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \cosh^{-1} \frac x a
+      | r = \map \ln {\frac x a + \sqrt {\paren {\frac x a}^2 - 1} }
+      | c = {{Defof|Inverse Hyperbolic Cosine/Real|index = 2|Real Inverse Hyperbolic Cosine}}
+}}
+{{eqn | r = \map \ln {\frac x a + \sqrt {\frac {x^2 - a^2} {a^2} } }
+      | c = 
+}}
+{{eqn | r = \map \ln {\frac x a + \frac {\sqrt {x^2 - a^2} } a}
+      | c = 
+}}
+{{eqn | r = \map \ln {\frac {x + \sqrt {x^2 - a^2} } a}
+      | c = 
+}}
+{{eqn | r = \map \ln {x + \sqrt {x^2 - a^2} } - \ln a
+      | c = [[Difference of Logarithms]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Inverse Hyperbolic Tangent of x over a in Logarithm Form}
+Tags: Inverse Hyperbolic Tangent
+
+\begin{theorem}
+:$\tanh^{-1} \dfrac x a = \dfrac 1 2 \map \ln {\dfrac {a + x} {a - x} }$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \tanh^{-1} \frac x a
+      | r = \frac 1 2 \map \ln {\frac {1 + \frac x a} {1 - \frac x a} }
+      | c = {{Defof|Inverse Hyperbolic Tangent|subdef = Real|index = 2}}
+}}
+{{eqn | r = \frac 1 2 \map \ln {\frac {a + x} {a - x} }
+      | c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $a$
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Inverse Hyperbolic Cotangent of x over a in Logarithm Form}
+Tags: Inverse Hyperbolic Cotangent
+
+\begin{theorem}
+:$\coth^{-1} \dfrac x a = \dfrac 1 2 \map \ln {\dfrac {x + a} {x - a} }$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \coth^{-1} \frac x a
+      | r = \frac 1 2 \map \ln {\frac {\frac x a + 1} {\frac x a - 1} }
+      | c = {{Defof|Inverse Hyperbolic Cotangent|subdef = Real|index = 2}}
+}}
+{{eqn | r = \frac 1 2 \map \ln {\frac {x + a} {x - a} }
+      | c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $a$
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Inverse Hyperbolic Secant of x over a in Logarithm Form}
+Tags: Inverse Hyperbolic Secant
+
+\begin{theorem}
+:$\operatorname{sech}^{-1} \dfrac x a = \ln \left({\dfrac {a + \sqrt{a^2 - x^2} } x}\right)$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \operatorname{sech}^{-1} \frac x a
+      | r = \ln \left({\frac {1 + \sqrt{1 - \left({\frac x a}\right)^2} } {\frac x a} }\right)
+      | c = Definition of [[Definition:Inverse Hyperbolic Secant/Real/Definition 2|Inverse Hyperbolic secant]]
+}}
+{{eqn | r = \ln \left({\frac {a \left({1 + \sqrt{\dfrac {a^2 - x^2} {a^2} } }\right)} x}\right)
+      | c = 
+}}
+{{eqn | r = \ln \left({\frac {a \left({\dfrac {a + \sqrt{a^2 - x^2} } a}\right)} x}\right)
+      | c = 
+}}
+{{eqn | r = \ln \left({\frac {a + \sqrt{a^2 - x^2} } x}\right)
+      | c = 
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Inverse Hyperbolic Cosecant of x over a in Logarithm Form}
+Tags: Inverse Hyperbolic Cosecant
+
+\begin{theorem}
+:$\csch^{-1} \dfrac x a = \map \ln {\dfrac {a + \sqrt {a^2 + x^2} } x}$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \csch^{-1} \frac x a
+      | r = \map \ln {\frac {1 + \sqrt {1 + \paren {\frac x a}^2} } {\frac x a} }
+      | c = {{Defof|Inverse Hyperbolic Cosecant|subdef = Real|index = 2}}
+}}
+{{eqn | r = \map \ln {\frac {a \paren {1 + \sqrt {\dfrac {a^2 + x^2} {a^2} } } } x}
+      | c = 
+}}
+{{eqn | r = \map \ln {\frac {a \paren {\dfrac {a + \sqrt {a^2 + x^2} } a} } x}
+      | c = 
+}}
+{{eqn | r = \map \ln {\frac {a + \sqrt {a^2 + x^2} } x}
+      | c = 
+}}
+{{end-eqn}}
+{{qed}}
+{{expand|Work out what happens when $x$ is real such that $x < 0$}}
+\end{proof}<|endoftext|>
+\section{Primitive of Root of x squared plus a squared/Logarithm Form}
+Tags: Primitive of Root of x squared plus a squared
+
+\begin{theorem}
+:$\displaystyle \int \sqrt {x^2 + a^2} \rd x = \frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \map \ln {x + \sqrt {x^2 + a^2} } + C$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \int \sqrt {x^2 + a^2} \rd x
+      | r = \frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \sinh^{-1} \frac x a + C
+      | c = [[Primitive of Root of x squared plus a squared/Inverse Hyperbolic Sine Form|Primitive of $\sqrt {x^2 + a^2}$ in $\sinh^{-1}$ form]]
+}}
+{{eqn | r = \frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \paren {\map \ln {x + \sqrt {x^2 + a^2} } - \ln a} + C
+      | c = [[Inverse Hyperbolic Sine of x over a in Logarithm Form|$\sinh^{-1} \dfrac x a$ in Logarithm Form]]
+}}
+{{eqn | r = \frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \map \ln {x + \sqrt {x^2 + a^2} } - \frac {a^2 \ln a} 2 + C
+      | c = simplifying
+}}
+{{eqn | r = \frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \map \ln {x + \sqrt {x^2 + a^2} } + C
+      | c = subsuming $\dfrac {-a^2 \ln a} 2$ into the [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of Root of x squared plus a squared/Inverse Hyperbolic Sine Form}
+Tags: Primitive of Root of x squared plus a squared
+
+\begin{theorem}
+:$\displaystyle \int \sqrt {x^2 + a^2} \rd x = \frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \sinh^{-1} \frac x a + C$
+\end{theorem}
+
+\begin{proof}
+Let:
+{{begin-eqn}}
+{{eqn | l = x
+      | r = a \sinh \theta
+}}
+{{eqn | n = 1
+      | ll= \leadsto
+      | l = \frac {\d x} {\d \theta}
+      | r = a \cosh \theta
+      | c = [[Derivative of Hyperbolic Sine]]
+}}
+{{end-eqn}}
+Also:
+{{begin-eqn}}
+{{eqn | l = x
+      | r = a \sinh \theta
+}}
+{{eqn | ll= \leadsto
+      | l = x^2 + a^2
+      | r = a^2 \sinh^2 \theta + a^2
+      | c = 
+}}
+{{eqn | r = a^2 \paren {\sinh^2 \theta + 1}
+      | c = 
+}}
+{{eqn | r = a^2 \cosh^2 \theta
+      | c = [[Difference of Squares of Hyperbolic Cosine and Sine]]
+}}
+{{eqn | n = 2
+      | ll= \leadsto
+      | l = \sqrt {x^2 + a^2}
+      | r = a \cosh \theta
+      | c = 
+}}
+{{end-eqn}}
+and:
+{{begin-eqn}}
+{{eqn | l = x
+      | r = a \sinh \theta
+}}
+{{eqn | n = 3
+      | ll= \leadsto
+      | l = \theta
+      | r = \sinh^{-1} \frac x a
+      | c = {{Defof|Inverse Hyperbolic Sine/Real|index = 1|Real Inverse Hyperbolic Sine}}
+}}
+{{end-eqn}}
+Thus:
+{{begin-eqn}}
+{{eqn | l = \int \sqrt {x^2 + a^2} \rd x
+      | r = \int \sqrt {x^2 + a^2} \, a \cosh \theta \rd \theta
+      | c = [[Integration by Substitution]] from $(1)$
+}}
+{{eqn | r = \int a^2 \cosh^2 \theta \rd \theta
+      | c = substituting for $\sqrt {x^2 + a^2}$ from $(2)$
+}}
+{{eqn | r = a^2 \int \cosh^2 \theta \rd \theta
+      | c = [[Primitive of Constant Multiple of Function]]
+}}
+{{eqn | r = a^2 \frac {\sinh \theta \cosh \theta + \theta} 2 + C
+      | c = [[Primitive of Square of Hyperbolic Cosine Function/Corollary|Primitive of Square of Hyperbolic Cosine Function: Corollary]]
+}}
+{{eqn | r = \frac 1 2 a \sinh \theta a \cosh \theta + \frac {a^2 \theta} 2 + C
+      | c = rearranging
+}}
+{{eqn | r = \frac 1 2 x a \cosh \theta + \frac {a^2 \theta} 2 + C
+      | c = substituting $x = a \sinh \theta$
+}}
+{{eqn | r = \frac 1 2 x \sqrt {x^2 + a^2} + \frac {a^2 \theta} 2 + C
+      | c = substituting $\sqrt {x^2 + a^2} = a \cosh \theta$ from $(2)$
+}}
+{{eqn | r = \frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \sinh^{-1} \frac x a + C
+      | c = substituting for $\theta$ from $(3)$
+}}
+{{end-eqn}}
+{{qed}}
+{{proofread|I have never seen this offered as a possible solution, so is it possible it's not valid?}}
+[[Category:Primitive of Root of x squared plus a squared]]
+n0z8ljsog5exrhytu4dg6cplvl8togi
+\end{proof}<|endoftext|>
+\section{Primitive of x squared over Root of x squared plus a squared/Inverse Hyperbolic Sine Form}
+Tags: Primitive of x squared over Root of x squared plus a squared
+
+\begin{theorem}
+:$\displaystyle \int \frac {x^2 \rd x} {\sqrt {x^2 + a^2} } = \frac {x \sqrt {x^2 + a^2} } 2 - \frac {a^2} 2 \sinh^{-1} \frac x a + C$
+\end{theorem}
+
+\begin{proof}
+With a view to expressing the problem in the form:
+:$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
+let:
+{{begin-eqn}}
+{{eqn | l = u
+      | r = x
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = \frac {\d u} {\d x}
+      | r = 1
+      | c = [[Power Rule for Derivatives]]
+}}
+{{end-eqn}}
+and let:
+{{begin-eqn}}
+{{eqn | l = \frac {\d v} {\d x}
+      | r = \frac x {\sqrt {x^2 + a^2} }
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = v
+      | r = \sqrt {x^2 + a^2}
+      | c = [[Primitive of x over Root of x squared plus a squared|Primitive of $\dfrac x {\sqrt {x^2 + a^2} }$]]
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = \int \frac {x^2 \rd x} {\sqrt {x^2 + a^2} }
+      | r = \int x \frac {x \rd x} {\sqrt {x^2 + a^2} }
+      | c = 
+}}
+{{eqn | r = x \sqrt {x^2 + a^2} - \int \sqrt {x^2 + a^2} \rd x
+      | c = [[Integration by Parts]]
+}}
+{{eqn | r = x \sqrt {x^2 + a^2} - \left({\frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \sinh^{-1} \frac x a}\right) + C
+      | c = [[Primitive of Root of x squared plus a squared/Inverse Hyperbolic Sine Form|Primitive of $\sqrt {x^2 + a^2}$]]
+}}
+{{eqn | r = \frac {x \sqrt {x^2 + a^2} } 2 - \frac {a^2} 2 \sinh^{-1} \frac x a + C
+      | c = 
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Primitive of x squared over Root of x squared plus a squared]]
+8e58ojdzgfurhum4175e1f5pzzq6ch0
+\end{proof}<|endoftext|>
+\section{Primitive of x squared over Root of x squared minus a squared/Logarithm Form}
+Tags: Primitive of x squared over Root of x squared minus a squared
+
+\begin{theorem}
+:$\displaystyle \int \frac {x^2 \rd x} {\sqrt {x^2 - a^2} } = \frac {x \sqrt {x^2 - a^2} } 2 + \frac {a^2} 2 \ln \size {x + \sqrt {x^2 - a^2} } + C$
+\end{theorem}
+
+\begin{proof}
+With a view to expressing the problem in the form:
+:$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
+let:
+{{begin-eqn}}
+{{eqn | l = u
+      | r = x
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = \frac {\d u} {\d x}
+      | r = 1
+      | c = [[Power Rule for Derivatives]]
+}}
+{{end-eqn}}
+and let:
+{{begin-eqn}}
+{{eqn | l = \frac {\d v} {\d x}
+      | r = \frac x {\sqrt {x^2 - a^2} }
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = v
+      | r = \sqrt {x^2 - a^2}
+      | c = [[Primitive of x over Root of x squared minus a squared|Primitive of $\dfrac x {\sqrt {x^2 - a^2} }$]]
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = \int \frac {x^2 \rd x} {\sqrt {x^2 - a^2} }
+      | r = \int x \frac {x \rd x} {\sqrt {x^2 - a^2} }
+      | c = 
+}}
+{{eqn | r = x \sqrt {x^2 - a^2} - \int \sqrt {x^2 - a^2} \rd x
+      | c = [[Integration by Parts]]
+}}
+{{eqn | r = x \sqrt {x^2 - a^2} - \paren {\frac {x \sqrt {x^2 - a^2} } 2 - \frac {a^2} 2 \ln \size {x + \sqrt {x^2 - a^2} } } + C
+      | c = [[Primitive of Root of x squared minus a squared/Logarithm Form|Primitive of $\sqrt {x^2 - a^2}$]]
+}}
+{{eqn | r = \frac {x \sqrt {x^2 - a^2} } 2 + \frac {a^2} 2 \ln \size {x + \sqrt {x^2 - a^2} } + C
+      | c = 
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of Root of x squared minus a squared/Inverse Hyperbolic Cosine Form}
+Tags: Primitive of Root of x squared minus a squared
+
+\begin{theorem}
+:$\displaystyle \int \sqrt {x^2 - a^2} \rd x = \frac {x \sqrt {x^2 - a^2} } 2 - \frac {a^2} 2 \cosh^{-1} \frac x a + C$
+\end{theorem}
+
+\begin{proof}
+Let:
+{{begin-eqn}}
+{{eqn | l = x
+      | r = a \cosh \theta
+}}
+{{eqn | n = 1
+      | ll= \leadsto
+      | l = \frac {\d x} {\d \theta}
+      | r = a \sinh \theta
+      | c = [[Derivative of Hyperbolic Cosine]]
+}}
+{{end-eqn}}
+Also:
+{{begin-eqn}}
+{{eqn | l = x
+      | r = a \cosh \theta
+}}
+{{eqn | ll= \leadsto
+      | l = x^2 - a^2
+      | r = a^2 \cosh^2 \theta - a^2
+      | c = 
+}}
+{{eqn | r = a^2 \paren {\cosh^2 \theta + 1}
+      | c = 
+}}
+{{eqn | r = a^2 \sinh^2 \theta
+      | c = [[Difference of Squares of Hyperbolic Cosine and Sine]]
+}}
+{{eqn | n = 2
+      | ll= \leadsto
+      | l = \sqrt {x^2 - a^2}
+      | r = a \sinh \theta
+      | c = 
+}}
+{{end-eqn}}
+and:
+{{begin-eqn}}
+{{eqn | l = x
+      | r = a \cosh \theta
+}}
+{{eqn | n = 3
+      | ll= \leadsto
+      | l = \theta
+      | r = \cosh^{-1} \frac x a
+      | c = {{Defof|Inverse Hyperbolic Cosine/Real|index = 1|Real Inverse Hyperbolic Cosine}}
+}}
+{{end-eqn}}
+Thus:
+{{begin-eqn}}
+{{eqn | l = \int \sqrt {x^2 - a^2} \rd x
+      | r = \int \sqrt {x^2 - a^2} \, a \sinh \theta \rd \theta
+      | c = [[Integration by Substitution]] from $(1)$
+}}
+{{eqn | r = \int a^2 \sinh^2 \theta \rd \theta
+      | c = substituting for $\sqrt {x^2 - a^2}$ from $(2)$
+}}
+{{eqn | r = a^2 \int \sinh^2 \theta \rd \theta
+      | c = [[Primitive of Constant Multiple of Function]]
+}}
+{{eqn | r = a^2 \frac {\sinh \theta \cosh \theta - \theta} 2 + C
+      | c = [[Primitive of Square of Hyperbolic Sine Function/Corollary|Primitive of $\sinh^2 \theta$: Corollary]]
+}}
+{{eqn | r = \frac 1 2 a \sinh \theta a \cosh \theta - \frac {a^2 \theta} 2 + C
+      | c = rearranging
+}}
+{{eqn | r = \frac 1 2 x a \sinh \theta - \frac {a^2 \theta} 2 + C
+      | c = substituting $x = a \cosh \theta$
+}}
+{{eqn | r = \frac 1 2 x \sqrt {x^2 - a^2} - \frac {a^2 \theta} 2 + C
+      | c = substituting $\sqrt {x^2 - a^2} = a \sinh \theta$ from $(2)$
+}}
+{{eqn | r = \frac {x \sqrt {x^2 - a^2} } 2 - \frac {a^2} 2 \cosh^{-1} \frac x a + C
+      | c = substituting for $\theta$ from $(3)$
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Primitive of Root of x squared minus a squared]]
+djyd129bzuhwpu94dqdp4jxwkiutkh4
+\end{proof}<|endoftext|>
+\section{Primitive of Root of x squared minus a squared/Logarithm Form}
+Tags: Primitive of Root of x squared minus a squared
+
+\begin{theorem}
+:$\displaystyle \int \sqrt {x^2 - a^2} \rd x = \frac {x \sqrt {x^2 - a^2} } 2 - \frac {a^2} 2 \ln \size {x + \sqrt {x^2 - a^2} } + C$
+\end{theorem}
+
+\begin{proof}
+We have that $\sqrt {x^2 - a^2}$ is defined only when $x^2 \ge a^2$, that is, either:
+:$x \ge a$
+or:
+:$x \le -a$
+where it is assumed that $a > 0$.
+First let $x \ge a$.
+{{begin-eqn}}
+{{eqn | l = x
+      | r = a \cosh u
+}}
+{{eqn | n = 1
+      | ll= \leadsto
+      | l = \frac {\d x} {\d u}
+      | r = a \sinh u
+      | c = [[Derivative of Hyperbolic Cosine]]
+}}
+{{end-eqn}}
+Also:
+{{begin-eqn}}
+{{eqn | l = x
+      | r = a \cosh u
+}}
+{{eqn | ll= \leadsto
+      | l = x^2 - a^2
+      | r = a^2 \cosh^2 u - a^2
+      | c = 
+}}
+{{eqn | r = a^2 \paren {\cosh^2 u + 1}
+      | c = 
+}}
+{{eqn | r = a^2 \sinh^2 u
+      | c = [[Difference of Squares of Hyperbolic Cosine and Sine]]
+}}
+{{eqn | n = 2
+      | ll= \leadsto
+      | l = \sqrt {x^2 - a^2}
+      | r = a \sinh u
+      | c = 
+}}
+{{end-eqn}}
+and:
+{{begin-eqn}}
+{{eqn | l = x
+      | r = a \cosh u
+}}
+{{eqn | n = 3
+      | ll= \leadsto
+      | l = u
+      | r = \cosh^{-1} \frac x a
+      | c = {{Defof|Inverse Hyperbolic Cosine/Real|index = 1|Real Inverse Hyperbolic Cosine}}
+}}
+{{end-eqn}}
+Thus:
+{{begin-eqn}}
+{{eqn | l = \int \sqrt {x^2 - a^2} \rd x
+      | r = \int \sqrt {x^2 - a^2} \, a \sinh u \rd u
+      | c = [[Integration by Substitution]] from $(1)$
+}}
+{{eqn | r = \int a^2 \sinh^2 u \rd u
+      | c = substituting for $\sqrt {x^2 - a^2}$ from $(2)$
+}}
+{{eqn | r = a^2 \int \sinh^2 u \rd u
+      | c = [[Primitive of Constant Multiple of Function]]
+}}
+{{eqn | r = a^2 \frac {\sinh u \cosh u - u} 2 + C
+      | c = [[Primitive of Square of Hyperbolic Sine Function/Corollary|Primitive of $\sinh^2 u$: Corollary]]
+}}
+{{eqn | r = \frac 1 2 a \sinh u a \cosh u - \frac {a^2 u} 2 + C
+      | c = rearranging
+}}
+{{eqn | r = \frac 1 2 x a \sinh u - \frac {a^2 u} 2 + C
+      | c = substituting $x = a \cosh u$
+}}
+{{eqn | r = \frac 1 2 x \sqrt {x^2 - a^2} - \frac {a^2 u} 2 + C
+      | c = substituting $\sqrt {x^2 - a^2} = a \sinh u$ from $(2)$
+}}
+{{eqn | r = \frac {x \sqrt {x^2 - a^2} } 2 - \frac {a^2} 2 \cosh^{-1} \frac x a + C
+      | c = substituting for $\theta$ from $(3)$
+}}
+{{eqn | r = \frac {x \sqrt {x^2 - a^2} } 2 - \frac {a^2} 2 \paren {\map \ln {x + \sqrt {x^2 - a^2} } - \ln a} + C
+      | c = [[Inverse Hyperbolic Cosine of x over a in Logarithm Form|$\cosh^{-1} \dfrac x a$ in Logarithm Form]]
+}}
+{{eqn | r = \frac {x \sqrt {x^2 - a^2} } 2 - \frac {a^2} 2 \map \ln {x + \sqrt {x^2 - a^2} } + \frac {a^2} 2 \ln a + C
+      | c = simplifying
+}}
+{{eqn | r = \frac {x \sqrt {x^2 - a^2} } 2 - \frac {a^2} 2 \map \ln {x + \sqrt {x^2 - a^2} } + C
+      | c = subsuming $\dfrac {a^2} 2 \ln a$ into [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]
+}}
+{{eqn | r = \frac {x \sqrt {x^2 - a^2} } 2 - \frac {a^2} 2 \ln \size {x + \sqrt {x^2 - a^2} } + C
+      | c = {{Defof |Absolute Value}}
+}}
+{{end-eqn}}
+Now suppose $x \le -a$.
+Let $z = -x$.
+Then:
+:$\d x = -\d z$
+and we then have:
+{{begin-eqn}}
+{{eqn | l = \int \sqrt {x^2 - a^2} \rd x
+      | r = -\int \sqrt {\paren {-z}^2 - a^2} \rd z
+      | c = [[Integration by Substitution]]
+}}
+{{eqn | r = -\int \sqrt {\paren z^2 - a^2} \rd z
+      | c = simplifying
+}}
+{{eqn | r = -\frac {z \sqrt {z^2 - a^2} } 2 + \frac {a^2} 2 \map \ln {z + \sqrt {z^2 - a^2} } + C
+      | c = from above
+}}
+{{eqn | r = -\frac {z \sqrt {z^2 - a^2} } 2 - \frac {a^2} 2 \paren {\map \ln {z - \sqrt {z^2 - a^2} } - \map \ln {a^2} } + C
+      | c = [[Negative of Logarithm of x plus Root x squared minus a squared|Negative of $\map \ln {z + \sqrt {z^2 - a^2} }$]]
+}}
+{{eqn | r = -\frac {z \sqrt {z^2 - a^2} } 2 - \frac {a^2} 2 \map \ln {z - \sqrt {z^2 - a^2} } + C
+      | c = subsuming $-\frac {a^2 \map \ln {a^2} } 2$ into [[Definition:Arbitrary Constant (Calculus)|constant]]
+}}
+{{eqn | r = -\frac {\paren {-x} \sqrt {\paren {-x}^2 - a^2} } 2 - \frac {a^2} 2 \map \ln {\paren {-x} - \sqrt {\paren {-x}^2 - a^2} } + C
+      | c = substituting back for $x$
+}}
+{{eqn | r = \frac {x \sqrt {x^2 - a^2} } 2 - \frac {a^2} 2 \map \ln {-x - \sqrt {x^2 - a^2} } + C
+      | c = simplifying
+}}
+{{eqn | r = \frac {x \sqrt {x^2 - a^2} } 2 - \frac {a^2} 2 \ln \size {x + \sqrt {x^2 - a^2} } + C
+      | c = as $-x - \sqrt {x^2 - a^2} > 0$: {{Defof |Absolute Value}}
+}}
+{{end-eqn}}
+The result follows.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of x squared over Root of x squared minus a squared/Inverse Hyperbolic Cosine Form}
+Tags: Primitive of x squared over Root of x squared minus a squared
+
+\begin{theorem}
+:$\displaystyle \int \frac {x^2 \rd x} {\sqrt {x^2 - a^2} } = \frac {x \sqrt {x^2 - a^2} } 2 + \frac {a^2} 2 \cosh^{-1} \frac x a + C$
+for $x > a$.
+\end{theorem}
+
+\begin{proof}
+With a view to expressing the problem in the form:
+:$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
+let:
+{{begin-eqn}}
+{{eqn | l = u
+      | r = x
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = \frac {\d u} {\d x}
+      | r = 1
+      | c = [[Power Rule for Derivatives]]
+}}
+{{end-eqn}}
+and let:
+{{begin-eqn}}
+{{eqn | l = \frac {\d v} {\d x}
+      | r = \frac x {\sqrt {x^2 - a^2} }
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = v
+      | r = \sqrt {x^2 - a^2}
+      | c = [[Primitive of x over Root of x squared minus a squared|Primitive of $\dfrac x {\sqrt {x^2 - a^2} }$]]
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = \int \frac {x^2 \rd x} {\sqrt {x^2 - a^2} }
+      | r = \int x \frac {x \rd x} {\sqrt {x^2 - a^2} }
+      | c = 
+}}
+{{eqn | r = x \sqrt {x^2 - a^2} - \int \sqrt {x^2 - a^2} \rd x
+      | c = [[Integration by Parts]]
+}}
+{{eqn | r = x \sqrt {x^2 - a^2} - \paren {\frac {x \sqrt {x^2 - a^2} } 2 - \frac {a^2} 2 \cosh^{-1} \frac x a} + C
+      | c = [[Primitive of Root of x squared minus a squared/Inverse Hyperbolic Cosine Form|Primitive of $\sqrt {x^2 - a^2}$]]
+}}
+{{eqn | r = \frac {x \sqrt {x^2 - a^2} } 2 + \frac {a^2} 2 \cosh^{-1} \frac x a + C
+      | c = simplifying
+}}
+{{end-eqn}}
+Note that because:
+:$\cosh^{-1} \frac x a$ is defined for $x \ge a$ only
+and:
+:$\dfrac {x^2} {\sqrt {x^2 - a^2} } is not defined for $x = a$
+$x$ is constrained as indicated.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of x by Inverse Hyperbolic Cosine of x over a}
+Tags: Primitives involving Inverse Hyperbolic Cosine Function
+
+\begin{theorem}
+:$\displaystyle \int x \cosh^{-1} \frac x a \rd x = \begin{cases}
+\paren {\dfrac {x^2} 2 - \dfrac {a^2} 4} \cosh^{-1} \dfrac x a - \dfrac {x \sqrt {x^2 - a^2} } 4 + C & : \cosh^{-1} \dfrac x a > 0 \\
+\paren {\dfrac {x^2} 2 - \dfrac {a^2} 4} \cosh^{-1} \dfrac x a + \dfrac {x \sqrt {x^2 - a^2} } 4 + C & : \cosh^{-1} \dfrac x a < 0 \end{cases}$
+\end{theorem}
+
+\begin{proof}
+With a view to expressing the [[Definition:Primitive|primitive]] in the form:
+:$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
+let:
+{{begin-eqn}}
+{{eqn | l = u
+      | r = \cosh^{-1} \frac x a
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = \frac {\d u} {\d x}
+      | r = \frac 1 {\sqrt {x^2 - a^2} }
+      | c = [[Derivative of Inverse Hyperbolic Cosine of x over a|Derivative of $\cosh^{-1} \dfrac x a$]]
+}}
+{{end-eqn}}
+and let:
+{{begin-eqn}}
+{{eqn | l = \frac {\d v} {\d x}
+      | r = x
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = v
+      | r = \frac {x^2} 2
+      | c = [[Primitive of Power]]
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = \int x \cosh^{-1} \frac x a \rd x
+      | r = \frac {x^2} 2 \cosh^{-1} \frac x a - \int \frac {x^2} 2 \paren {\frac 1 {\sqrt {x^2 - a^2} } } \rd x + C
+      | c = [[Integration by Parts]]
+}}
+{{eqn | r = \frac {x^2} 2 \cosh^{-1} \frac x a - \frac 1 2 \int \frac {x^2 \rd x} {\sqrt {x^2 - a^2} } + C
+      | c = simplifying
+}}
+{{eqn | r = \frac {x^2} 2 \cosh^{-1} \frac x a - \frac 1 2 \paren {\frac {x \sqrt {x^2 - a^2} } 2 + \frac {a^2} 2 \cosh^{-1} \frac x a} + C
+      | c = [[Primitive of x squared over Root of x squared minus a squared/Inverse Hyperbolic Cosine Form|Primitive of $\dfrac {x^2} {\sqrt {x^2 - a^2} }$]]
+}}
+{{eqn | r = \paren {\frac {x^2} 2 - \frac {a^2} 4} \cosh^{-1} \frac x a - \frac {x \sqrt {x^2 - a^2} } 4 + C
+      | c = simplifying
+}}
+{{end-eqn}}
+{{qed}}
+{{finish|Analysis to be done for the negative case. The confusion here is that $\cosh^{-1} \frac x a$ is ''defined'' as being positive.}}
+\end{proof}<|endoftext|>
+\section{Primitive of x squared over x squared minus a squared/Inverse Hyperbolic Cotangent Form}
+Tags: Primitive of x squared over x squared minus a squared
+
+\begin{theorem}
+:$\displaystyle \int \frac {x^2 \ \mathrm d x} {x^2 - a^2} = x - a \coth^{-1} \frac x a + C$
+for $x^2 > a^2$.
+\end{theorem}
+
+\begin{proof}
+Let:
+{{begin-eqn}}
+{{eqn | l = \int \frac {x^2 \ \mathrm d x} {x^2 - a^2}
+      | r = \int \frac {x^2 - a^2 + a^2} {x^2 - a^2} \ \mathrm d x
+      | c = 
+}}
+{{eqn | r = \int \frac {x^2 - a^2} {x^2 - a^2} \ \mathrm d x + \int \frac {a^2} {x^2 - a^2} \ \mathrm d x
+      | c = 
+}}
+{{eqn | r = \int \mathrm d x + a^2 \int \frac {\mathrm d x} {x^2 - a^2}
+      | c = [[Primitive of Constant Multiple of Function]]
+}}
+{{eqn | r = x + a^2 \int \frac {\mathrm d x} {x^2 - a^2} + C
+      | c = [[Primitive of Constant]]
+}}
+{{eqn | r = x + a^2 \left({-\frac 1 a \coth^{-1} {\frac x a} }\right) + C
+      | c = [[Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Cotangent Form|Primitive of Reciprocal of $x^2 - a^2$: $\coth^{-1}$ form]]
+}}
+{{eqn | r = x - a \coth^{-1} {\frac x a} + C
+      | c = simplifying
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Primitive of x squared over x squared minus a squared]]
+l98lpnwxoktlylc6pngqudn6q75wtyc
+\end{proof}<|endoftext|>
+\section{Primitive of x squared over a squared minus x squared/Logarithm Form}
+Tags: Primitive of x squared over a squared minus x squared
+
+\begin{theorem}
+:$\displaystyle \int \frac {x^2 \rd x} {a^2 - x^2} = -x + \frac a 2 \map \ln {\frac {a + x} {a - x} } + C$
+for $x^2 < a^2$.
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \int \frac {x^2 \rd x} {a^2 - x^2}
+      | r = \int \frac {x^2 - a^2 + a^2} {a^2 - x^2} \rd x
+      | c = 
+}}
+{{eqn | r = \int \frac {- \left({a^2 - x^2}\right)} {a^2 - x^2} \rd x + \int \frac {a^2} {a^2 - x^2} \rd x
+      | c = [[Linear Combination of Integrals]]
+}}
+{{eqn | r = - \int \d x + a^2 \int \frac {\d x} {a^2 - x^2}
+      | c = [[Linear Combination of Integrals]]
+}}
+{{eqn | r = -x + a^2 \int \frac {\d x} {a^2 - x^2} + C
+      | c = [[Primitive of Constant]]
+}}
+{{eqn | r = -x + a^2 \paren {\frac 1 {2 a} \map \ln {\frac {a + x} {a - x} } } + C
+      | c = [[Primitive of Reciprocal of a squared minus x squared/Logarithm Form|Primitive of $\dfrac 1 {a^2 - x^2}$: Logarithm Form]]
+}}
+{{eqn | r = -x + \frac a 2 \map \ln {\frac {a + x} {a - x} } + C
+      | c = simplifying
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of x squared over a squared minus x squared/Inverse Hyperbolic Tangent Form}
+Tags: Primitive of x squared over a squared minus x squared
+
+\begin{theorem}
+:$\displaystyle \int \frac {x^2 \rd x} {a^2 - x^2} = -x + a \tanh^{-1} \frac x a + C$
+for $x^2 < a^2$.
+\end{theorem}
+
+\begin{proof}
+Let:
+{{begin-eqn}}
+{{eqn | l = \int \frac {x^2 \rd x} {a^2 - x^2}
+      | r = \int \frac {x^2 - a^2 + a^2} {a^2 - x^2} \rd x
+      | c = 
+}}
+{{eqn | r = \int \frac {- \left({a^2 - x^2}\right)} {a^2 - x^2} \rd x + \int \frac {a^2} {a^2 - x^2} \rd x
+      | c = [[Linear Combination of Integrals]]
+}}
+{{eqn | r = - \int \rd x + a^2 \int \frac {\rd x} {a^2 - x^2}
+      | c = [[Linear Combination of Integrals]]
+}}
+{{eqn | r = -x + a^2 \int \frac {\rd x} {a^2 - x^2} + C
+      | c = [[Primitive of Constant]]
+}}
+{{eqn | r = -x + a^2 \left({\frac 1 a \tanh^{-1} \frac x a}\right) + C
+      | c = [[Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Tangent Form|Primitive of Reciprocal of $a^2 - x^2$: $\tanh^{-1}$ form]]
+}}
+{{eqn | r = -x + a \tanh^{-1} \frac x a + C
+      | c = simplifying
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Primitive of x squared over a squared minus x squared]]
+pg0zh6tf9dnyu8rdcx9ic2og1wdag3n
+\end{proof}<|endoftext|>
+\section{Primitive of x by Inverse Hyperbolic Tangent of x over a}
+Tags: Primitives involving Inverse Hyperbolic Tangent Function
+
+\begin{theorem}
+:$\displaystyle \int x \tanh^{-1} \frac x a \rd x = \frac {a x} 2 + \frac {x^2 - a^2} 2 \tanh^{-1} \frac x a + C$
+\end{theorem}
+
+\begin{proof}
+With a view to expressing the [[Definition:Primitive|primitive]] in the form:
+:$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
+let:
+{{begin-eqn}}
+{{eqn | l = u
+      | r = \tanh^{-1} \frac x a
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = \frac {\d u} {\d x}
+      | r = \frac a {a^2 - x^2}
+      | c = [[Derivative of Inverse Hyperbolic Tangent of x over a|Derivative of $\tanh^{-1} \dfrac x a$]]
+}}
+{{end-eqn}}
+and let:
+{{begin-eqn}}
+{{eqn | l = \frac {\mathrm d v} {\mathrm d x}
+      | r = x
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = v
+      | r = \frac {x^2} 2
+      | c = [[Primitive of Power]]
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = \int x \tanh^{-1} \frac x a \rd x
+      | r = \frac {x^2} 2 \tanh^{-1} \frac x a - \int \frac {x^2} 2 \paren {\frac a {a^2 - x^2} } \rd x + C
+      | c = [[Integration by Parts]]
+}}
+{{eqn | r = \frac {x^2} 2 \tanh^{-1} \frac x a - \frac a 2 \int \frac {x^2 \rd x} {a^2 - x^2} + C
+      | c = [[Primitive of Constant Multiple of Function]]
+}}
+{{eqn | r = \frac {x^2} 2 \tanh^{-1} \frac x a - \frac a 2 \paren {-x + a \tanh^{-1} \frac x a} + C
+      | c = [[Primitive of x squared over a squared minus x squared/Inverse Hyperbolic Tangent Form|Primitive of $\dfrac {x^2} {a^2 - x^2}$]]
+}}
+{{eqn | r = \frac {a x} 2 + \frac {x^2 - a^2} 2 \tanh^{-1} \frac x a + C
+      | c = simplifying
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of x by Inverse Hyperbolic Cotangent of x over a}
+Tags: Primitives involving Inverse Hyperbolic Cotangent Function
+
+\begin{theorem}
+:$\displaystyle \int x \coth^{-1} \frac x a \rd x = \frac {a x} 2 + \frac {x^2 - a^2} 2 \coth^{-1} \frac x a + C$
+\end{theorem}
+
+\begin{proof}
+With a view to expressing the [[Definition:Primitive|primitive]] in the form:
+:$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
+let:
+{{begin-eqn}}
+{{eqn | l = u
+      | r = \coth^{-1} \frac x a
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = \frac {\d u} {\d x}
+      | r = \frac a {a^2 - x^2}
+      | c = [[Derivative of Inverse Hyperbolic Cotangent of x over a|Derivative of $\coth^{-1} \dfrac x a$]]
+}}
+{{end-eqn}}
+and let:
+{{begin-eqn}}
+{{eqn | l = \frac {\d v} {\d x}
+      | r = x
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = v
+      | r = \frac {x^2} 2
+      | c = [[Primitive of Power]]
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = \int x \coth^{-1} \frac x a \rd x
+      | r = \frac {x^2} 2 \coth^{-1} \frac x a - \int \frac {x^2} 2 \paren {\frac a {a^2 - x^2} } \rd x + C
+      | c = [[Integration by Parts]]
+}}
+{{eqn | r = \frac {x^2} 2 \coth^{-1} \frac x a + \frac a 2 \int \frac {x^2 \rd x} {x^2 - a^2} + C
+      | c = [[Primitive of Constant Multiple of Function]]
+}}
+{{eqn | r = \frac {x^2} 2 \coth^{-1} \frac x a + \frac a 2 \paren {x - a \coth^{-1} \frac x a} + C
+      | c = [[Primitive of x squared over x squared minus a squared/Inverse Hyperbolic Cotangent Form|Primitive of $\dfrac {x^2} {x^2 - a^2}$]]
+}}
+{{eqn | r = \frac {a x} 2 + \frac {x^2 - a^2} 2 \coth^{-1} \frac x a + C
+      | c = simplifying
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of x squared by Inverse Hyperbolic Sine of x over a}
+Tags: Primitives involving Inverse Hyperbolic Sine Function
+
+\begin{theorem}
+:$\displaystyle \int x^2 \sinh^{-1} \frac x a \rd x = \frac {x^3} 3 \sinh^{-1} \frac x a + \frac {\paren {2 a^2 - x^2} \sqrt {x^2 + a^2} } 9 + C$
+\end{theorem}
+
+\begin{proof}
+With a view to expressing the [[Definition:Primitive|primitive]] in the form:
+:$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
+let:
+{{begin-eqn}}
+{{eqn | l = u
+      | r = \sinh^{-1} \frac x a
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = \frac {\d u} {\d x}
+      | r = \frac 1 {\sqrt {x^2 + a^2} }
+      | c = [[Derivative of Inverse Hyperbolic Sine of x over a|Derivative of $\sinh^{-1} \dfrac x a$]]
+}}
+{{end-eqn}}
+and let:
+{{begin-eqn}}
+{{eqn | l = \frac {\d v} {\d x}
+      | r = x^2
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = v
+      | r = \frac {x^3} 3
+      | c = [[Primitive of Power]]
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = \int x^2 \sinh^{-1} \frac x a \rd x
+      | r = \frac {x^3} 3 \sinh^{-1} \frac x a - \int \frac {x^3} 3 \paren {\frac 1 {\sqrt {x^2 + a^2} } } \rd x + C
+      | c = [[Integration by Parts]]
+}}
+{{eqn | r = \frac {x^3} 3 \sinh^{-1} \frac x a - \frac 1 3 \int \frac {x^3 \rd x} {\sqrt {x^2 + a^2} } + C
+      | c = simplifying
+}}
+{{eqn | r = \frac {x^3} 3 \sinh^{-1} \frac x a - \frac 1 3 \paren {\frac {\paren {\sqrt {x^2 + a^2} }^3} 3 - a^2 \sqrt {x^2 + a^2} } + C
+      | c = [[Primitive of x cubed over Root of x squared plus a squared|Primitive of $\dfrac {x^3} {\sqrt {x^2 + a^2} }$]]
+}}
+{{eqn | r = \frac {x^3} 3 \sinh^{-1} \frac x a + \frac {\paren {2 a^2 - x^2} \sqrt {x^2 + a^2} } 9 + C
+      | c = simplifying
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of x squared by Inverse Hyperbolic Cosine of x over a}
+Tags: Primitives involving Inverse Hyperbolic Cosine Function
+
+\begin{theorem}
+:$\displaystyle \int x^2 \cosh^{-1} \frac x a \ \mathrm d x = \begin{cases}
+\displaystyle \frac {x^3} 3 \cosh^{-1} \frac x a - \frac {\left({x^2 + 2 a^2}\right) \sqrt {x^2 - a^2} } 9 + C & : \cosh^{-1} \frac x a > 0 \\
+\displaystyle \frac {x^3} 3 \cosh^{-1} \frac x a - \frac {\left({x^2 + 2 a^2}\right) \sqrt {x^2 - a^2} } 9 + C & : \cosh^{-1} \frac x a < 0
+\end{cases}$
+\end{theorem}
+
+\begin{proof}
+With a view to expressing the [[Definition:Primitive|primitive]] in the form:
+:$\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$
+let:
+{{begin-eqn}}
+{{eqn | l = u
+      | r = \cosh^{-1} \frac x a
+      | c = 
+}}
+{{eqn | ll= \implies
+      | l = \frac {\mathrm d u} {\mathrm d x}
+      | r = \frac 1 {\sqrt {x^2 - a^2} }
+      | c = [[Derivative of Inverse Hyperbolic Cosine of x over a|Derivative of $\cosh^{-1} \dfrac x a$]]
+}}
+{{end-eqn}}
+and let:
+{{begin-eqn}}
+{{eqn | l = \frac {\mathrm d v} {\mathrm d x}
+      | r = x^2
+      | c = 
+}}
+{{eqn | ll= \implies
+      | l = v
+      | r = \frac {x^3} 3
+      | c = [[Primitive of Power]]
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = \int x^2 \cosh^{-1} \frac x a \ \mathrm d x
+      | r = \frac {x^3} 3 \cosh^{-1} \frac x a - \int \frac {x^3} 3 \left({\frac 1 {\sqrt {x^2 - a^2} } }\right) \ \mathrm d x + C
+      | c = [[Integration by Parts]]
+}}
+{{eqn | r = \frac {x^3} 3 \cosh^{-1} \frac x a - \frac 1 3 \int \frac {x^3 \ \mathrm d x} {\sqrt {x^2 - a^2} } + C
+      | c = simplifying
+}}
+{{eqn | r = \frac {x^3} 3 \cosh^{-1} \frac x a - \frac 1 3 \left({\frac {\left({\sqrt {x^2 - a^2} }\right)^3} 3 + a^2 \sqrt {x^2 - a^2} }\right) + C
+      | c = [[Primitive of x cubed over Root of x squared minus a squared|Primitive of $\dfrac {x^3} {\sqrt {x^2 - a^2} }$]]
+}}
+{{eqn | r = \frac {x^3} 3 \cosh^{-1} \frac x a - \frac {\left({x^2 + 2 a^2}\right) \sqrt {x^2 - a^2} } 9 + C
+      | c = simplifying
+}}
+{{end-eqn}}
+{{qed}}
+{{finish|Analysis to be done for the negative case. The confusion here is that $\cosh^{-1} \frac x a$ is ''defined'' as being positive.}}
+\end{proof}<|endoftext|>
+\section{Primitive of x squared by Inverse Hyperbolic Tangent of x over a}
+Tags: Primitives involving Inverse Hyperbolic Tangent Function
+
+\begin{theorem}
+:$\displaystyle \int x^2 \tanh^{-1} \frac x a \ \mathrm d x = \frac {a x^2} 6 + \frac {x^3} 3 \tanh^{-1} \frac x a + \frac {a^3} 6 \ln \left({a^2 - x^2}\right) + C$
+\end{theorem}
+
+\begin{proof}
+With a view to expressing the [[Definition:Primitive|primitive]] in the form:
+:$\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$
+let:
+{{begin-eqn}}
+{{eqn | l = u
+      | r = \tanh^{-1} \frac x a
+      | c = 
+}}
+{{eqn | ll= \implies
+      | l = \frac {\mathrm d u} {\mathrm d x}
+      | r = \frac a {a^2 - x^2}
+      | c = [[Derivative of Inverse Hyperbolic Tangent of x over a|Derivative of $\tanh^{-1} \dfrac x a$]]
+}}
+{{end-eqn}}
+and let:
+{{begin-eqn}}
+{{eqn | l = \frac {\mathrm d v} {\mathrm d x}
+      | r = x^2
+      | c = 
+}}
+{{eqn | ll= \implies
+      | l = v
+      | r = \frac {x^3} 3
+      | c = [[Primitive of Power]]
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = \int x^2 \tanh^{-1} \frac x a \ \mathrm d x
+      | r = \frac {x^3} 3 \tanh^{-1} \frac x a - \int \frac {x^3} 3 \left({\frac a {a^2 - x^2} }\right) \ \mathrm d x + C
+      | c = [[Integration by Parts]]
+}}
+{{eqn | r = \frac {x^3} 3 \tanh^{-1} \frac x a - \frac a 3 \int \frac {x^3 \ \mathrm d x} {a^2 - x^2} + C
+      | c = simplifying
+}}
+{{eqn | r = \frac {x^3} 3 \tanh^{-1} \frac x a - \frac a 3 \left({-\frac {x^2} 2 - \frac {a^2} 2 \ln \left({a^2 - x^2}\right) }\right) + C
+      | c = [[Primitive of x cubed over a squared minus x squared|Primitive of $\dfrac {x^3} {a^2 - x^2}$]]
+}}
+{{eqn | r = \frac {a x^2} 6 + \frac {x^3} 3 \tanh^{-1} \frac x a + \frac {a^3} 6 \ln \left({a^2 - x^2}\right) + C
+      | c = simplifying
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of x squared by Inverse Hyperbolic Cotangent of x over a}
+Tags: Primitives involving Inverse Hyperbolic Cotangent Function
+
+\begin{theorem}
+:$\displaystyle \int x^2 \coth^{-1} \frac x a \ \mathrm d x = \frac {a x^2} 6 + \frac {x^3} 3 \coth^{-1} \frac x a + \frac {a^3} 6 \ln \left({x^2 - a^2}\right) + C$
+\end{theorem}
+
+\begin{proof}
+With a view to expressing the [[Definition:Primitive|primitive]] in the form:
+:$\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$
+let:
+{{begin-eqn}}
+{{eqn | l = u
+      | r = \coth^{-1} \frac x a
+      | c = 
+}}
+{{eqn | ll= \implies
+      | l = \frac {\mathrm d u} {\mathrm d x}
+      | r = \frac {-a} {x^2 - a^2}
+      | c = [[Derivative of Inverse Hyperbolic Cotangent of x over a|Derivative of $\coth^{-1} \dfrac x a$]]
+}}
+{{end-eqn}}
+and let:
+{{begin-eqn}}
+{{eqn | l = \frac {\mathrm d v} {\mathrm d x}
+      | r = x^2
+      | c = 
+}}
+{{eqn | ll= \implies
+      | l = v
+      | r = \frac {x^3} 3
+      | c = [[Primitive of Power]]
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = \int x^2 \coth^{-1} \frac x a \ \mathrm d x
+      | r = \frac {x^3} 3 \coth^{-1} \frac x a - \int \frac {x^3} 3 \left({\frac {-a} {x^2 - a^2} }\right) \ \mathrm d x + C
+      | c = [[Integration by Parts]]
+}}
+{{eqn | r = \frac {x^3} 3 \coth^{-1} \frac x a + \frac a 3 \int \frac {x^3 \ \mathrm d x} {x^2 - a^2} + C
+      | c = simplifying
+}}
+{{eqn | r = \frac {x^3} 3 \coth^{-1} \frac x a + \frac a 3 \left({\frac {x^2} 2 + \frac {a^2} 2 \ln \left({x^2 - a^2}\right) }\right) + C
+      | c = [[Primitive of x cubed over x squared minus a squared|Primitive of $\dfrac {x^3} {x^2 - a^2}$]]
+}}
+{{eqn | r = \frac {a x^2} 6 + \frac {x^3} 3 \coth^{-1} \frac x a + \frac {a^3} 6 \ln \left({x^2 - a^2}\right) + C
+      | c = simplifying
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of Inverse Hyperbolic Sine of x over a over x}
+Tags: Primitives involving Inverse Hyperbolic Sine Function
+
+\begin{theorem}
+:$\displaystyle \int \frac {\sinh^{-1} \dfrac x a \ \mathrm d x} x = \begin{cases}
+\displaystyle \sum_{k \mathop \ge 0} \frac {\left({-1}\right)^k \left({2 k + 1}\right)!} {2^{2 k} \left({k!}\right)^2 \left({2 k + 1}\right)^3} \left({\frac x a}\right)^{2 k + 1} + C & : \left\vert{x}\right\vert < a \\
+\displaystyle \frac {\ln^2 \left({\dfrac {2 x} a}\right)} 2 + \sum_{k \mathop \ge 0} \frac {\left({-1}\right)^k \left({2 k + 1}\right)!} {2^{2 k} \left({k!}\right)^2 \left({2 k + 1}\right)^3 \left({2 k}\right)^2} \left({\frac a x}\right)^{2 k} + C & : x > a \\
+\displaystyle \frac {-\ln^2 \left({\dfrac {2 x} a}\right)} 2 + \sum_{k \mathop \ge 0} \frac {\left({-1}\right)^{k + 1} \left({2 k + 1}\right)!} {2^{2 k} \left({k!}\right)^2 \left({2 k + 1}\right)^3 \left({2 k}\right)^2} \left({\frac a x}\right)^{2 k} + C & : x < a
+\end{cases}$
+\end{theorem}<|endoftext|>
+\section{Primitive of Inverse Hyperbolic Cosine of x over a over x}
+Tags: Primitives involving Inverse Hyperbolic Cosine Function
+
+\begin{theorem}
+:$\displaystyle \int \frac {\cosh^{-1} \dfrac x a \ \mathrm d x} x = \begin{cases}
+\displaystyle \frac {\ln^2 \left({\dfrac {2 x} a}\right)} 2 + \sum_{k \mathop \ge 0} \frac {\left({2 k + 1}\right)!} {2^{2 k} \left({k!}\right)^2 \left({2 k + 1}\right)^3 \left({2 k}\right)^2} \left({\frac a x}\right)^{2 k} + C & : \cosh^{-1} \dfrac x a > 0 \\
+\displaystyle \frac {-\ln^2 \left({\dfrac {2 x} a}\right)} 2 - \sum_{k \mathop \ge 0} \frac {\left({2 k + 1}\right)!} {2^{2 k} \left({k!}\right)^2 \left({2 k + 1}\right)^3 \left({2 k}\right)^2} \left({\frac a x}\right)^{2 k} + C & : \cosh^{-1} \dfrac x a < 0 \\
+\end{cases}$
+\end{theorem}<|endoftext|>
+\section{Primitive of Inverse Hyperbolic Tangent of x over a over x}
+Tags: Primitives involving Inverse Hyperbolic Tangent Function
+
+\begin{theorem}
+:$\ds \int \frac {\tanh^{-1} \dfrac x a \rd x} x = \sum_{k \mathop \ge 0} \frac 1 {\paren {2 k + 1}^2} \paren {\frac x a}^{2 k + 1}$
+\end{theorem}
+
+\begin{proof}
+{{proof wanted|Taylor expansion, and integrate term by term}}
+\end{proof}<|endoftext|>
+\section{Primitive of Inverse Hyperbolic Cotangent of x over a over x}
+Tags: Primitives involving Inverse Hyperbolic Cotangent Function
+
+\begin{theorem}
+:$\displaystyle \int \frac {\coth^{-1} \dfrac x a \rd x} x = -\sum_{k \mathop \ge 0} \frac 1 {\paren {2 k + 1}^2} \paren {\frac a x}^{2 k + 1}$
+\end{theorem}<|endoftext|>
+\section{Primitive of Inverse Hyperbolic Secant of x over a over x}
+Tags: Primitives involving Inverse Hyperbolic Secant Function
+
+\begin{theorem}
+:$\displaystyle \int \frac {\sech^{-1} \dfrac x a \rd x} x = \begin {cases}
+\displaystyle -\frac {\map \ln {\dfrac a x} \map \ln {\dfrac {4 a} x} } 2 - \sum_{k \mathop \ge 0} \frac {\paren {2 k + 1}!} {2^{2 k} \paren {k!}^2 \paren {2 k + 1}^3 \paren {2 k}^2} \paren {\frac x a}^{2 k} + C & : \sech^{-1} \dfrac x a > 0 \\
+\displaystyle \frac {\map \ln {\dfrac a x} \map \ln {\dfrac {4 a} x} } 2 + \sum_{k \mathop \ge 0} \frac {\paren {2 k + 1}!} {2^{2 k} \paren {k!}^2 \paren {2 k + 1}^3 \paren {2 k}^2} \paren {\frac x a}^{2 k} + C & : \sech^{-1} \dfrac x a < 0 \\
+\end {cases}$
+\end{theorem}<|endoftext|>
+\section{Primitive of Inverse Hyperbolic Cosecant of x over a over x}
+Tags: Primitives involving Inverse Hyperbolic Cosecant Function
+
+\begin{theorem}
+:$\displaystyle \int \frac {\operatorname{csch}^{-1} \dfrac x a \ \mathrm d x} x = \begin{cases}
+\displaystyle \frac {\ln \left({\dfrac x a}\right) \ln \left({\dfrac {4 a} x}\right)} 2 - \sum_{k \mathop \ge 0} \frac {\left({2 k + 1}\right)!} {2^{2 k} \left({k!}\right)^2 \left({2 k + 1}\right)^3 \left({2 k}\right)^2} \left({\frac x a}\right)^{2 k} + C & : 0 < x < a \\
+\displaystyle \frac {\ln \left({\dfrac {-x} a}\right) \ln \left({\dfrac {-x} {4 a} }\right)} 2 + \sum_{k \mathop \ge 0} \frac {\left({2 k + 1}\right)!} {2^{2 k} \left({k!}\right)^2 \left({2 k + 1}\right)^3 \left({2 k}\right)^2} \left({\frac x a}\right)^{2 k} + C & : -a < x < 0 \\
+\displaystyle \sum_{k \mathop \ge 0} \frac {\left({-1}\right)^{k + 1} \left({2 k + 1}\right)!} {2^{2 k} \left({k!}\right)^2 \left({2 k + 1}\right)^3} \left({\frac a x}\right)^{2 k + 1} + C & : \left\vert{x}\right\vert > a \\
+\end{cases}$
+\end{theorem}<|endoftext|>
+\section{Primitive of Inverse Hyperbolic Sine of x over a over x squared}
+Tags: Primitives involving Inverse Hyperbolic Sine Function
+
+\begin{theorem}
+:$\displaystyle \int \frac {\sinh^{-1} \dfrac x a \ \mathrm d x} {x^2} = \frac {-\sinh^{-1} \dfrac x a} x - \frac 1 a \ln \left({\frac {a + \sqrt {x^2 + a^2} } x}\right)$
+\end{theorem}
+
+\begin{proof}
+With a view to expressing the [[Definition:Primitive|primitive]] in the form:
+:$\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$
+let:
+{{begin-eqn}}
+{{eqn | l = u
+      | r = \sinh^{-1} \frac x a
+      | c = 
+}}
+{{eqn | ll= \implies
+      | l = \frac {\mathrm d u} {\mathrm d x}
+      | r = \frac 1 {\sqrt {x^2 + a^2} }
+      | c = [[Derivative of Inverse Hyperbolic Sine of x over a|Derivative of $\sinh^{-1} \dfrac x a$]]
+}}
+{{end-eqn}}
+and let:
+{{begin-eqn}}
+{{eqn | l = \frac {\mathrm d v} {\mathrm d x}
+      | r = \frac 1 {x^2}
+      | c = 
+}}
+{{eqn | ll= \implies
+      | l = v
+      | r = \frac {-1} x
+      | c = [[Primitive of Power]]
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = \int \frac {\sinh^{-1} \dfrac x a \ \mathrm d x} {x^2}
+      | r = \left({\sinh^{-1} \frac x a}\right) \left({\frac {-1} x}\right) - \int \left({\frac {-1} x}\right) \left({\frac 1 {\sqrt {x^2 + a^2} } }\right) \ \mathrm d x + C
+      | c = [[Integration by Parts]]
+}}
+{{eqn | r = \frac {-\sinh^{-1} \dfrac x a} x + \int \frac {\mathrm d x} {x \sqrt {x^2 + a^2} } + C
+      | c = simplifying
+}}
+{{eqn | r = \frac {-\sinh^{-1} \dfrac x a} x - \frac 1 a \ln \left({\frac {a + \sqrt {x^2 + a^2} } x}\right) + C
+      | c = [[Primitive of Reciprocal of x by Root of x squared plus a squared/Logarithm Form|Primitive of $\dfrac 1 {x \sqrt {x^2 + a^2} }$]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of Inverse Hyperbolic Cosine of x over a over x squared}
+Tags: Primitives involving Inverse Hyperbolic Cosine Function
+
+\begin{theorem}
+:$\displaystyle \int \frac {\cosh^{-1} \dfrac x a \ \mathrm d x} {x^2} = \begin{cases}
+\displaystyle \frac {-\cosh^{-1} \dfrac x a} x - \frac 1 a \ln \left({\frac {a + \sqrt {x^2 + a^2} } x}\right) & : \cosh^{-1} \dfrac x a > 0 \\
+\displaystyle \frac {-\cosh^{-1} \dfrac x a} x + \frac 1 a \ln \left({\frac {a + \sqrt {x^2 + a^2} } x}\right) & : \cosh^{-1} \dfrac x a < 0 \\
+\end{cases}$
+\end{theorem}
+
+\begin{proof}
+With a view to expressing the [[Definition:Primitive|primitive]] in the form:
+:$\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$
+let:
+{{begin-eqn}}
+{{eqn | l = u
+      | r = \cosh^{-1} \frac x a
+      | c = 
+}}
+{{eqn | ll= \implies
+      | l = \frac {\mathrm d u} {\mathrm d x}
+      | r = \frac 1 {\sqrt {x^2 - a^2} }
+      | c = [[Derivative of Inverse Hyperbolic Cosine of x over a|Derivative of $\cosh^{-1} \dfrac x a$]]
+}}
+{{end-eqn}}
+and let:
+{{begin-eqn}}
+{{eqn | l = \frac {\mathrm d v} {\mathrm d x}
+      | r = \frac 1 {x^2}
+      | c = 
+}}
+{{eqn | ll= \implies
+      | l = v
+      | r = \frac {-1} x
+      | c = [[Primitive of Power]]
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = \int \frac {\cosh^{-1} \dfrac x a \ \mathrm d x} {x^2}
+      | r = \left({\cosh^{-1} \frac x a}\right) \left({\frac {-1} x}\right) - \int \left({\frac {-1} x}\right) \left({\frac 1 {\sqrt {x^2 - a^2} } }\right) \ \mathrm d x + C
+      | c = [[Integration by Parts]]
+}}
+{{eqn | r = \frac {-\cosh^{-1} \dfrac x a} x + \int \frac {\mathrm d x} {x \sqrt {x^2 - a^2} } + C
+      | c = simplifying
+}}
+{{eqn | r = \frac {-\cosh^{-1} \dfrac x a} x - \frac 1 a \operatorname{arcsec} \left\vert{\frac x a}\right\vert + C
+      | c = [[Primitive of Reciprocal of x by Root of x squared minus a squared|Primitive of $\dfrac 1 {x \sqrt {x^2 - a^2} }$]]
+}}
+{{end-eqn}}
+{{qed}}
+{{finish|Not sure how to get there from here, I wonder if there is a problem with the required target}}
+\end{proof}<|endoftext|>
+\section{Primitive of Inverse Hyperbolic Tangent of x over a over x squared}
+Tags: Primitives involving Inverse Hyperbolic Tangent Function
+
+\begin{theorem}
+:$\ds \int \frac {\tanh^{-1} \dfrac x a \rd x} {x^2} = \frac {-\tanh^{-1} \dfrac x a} x + \frac 1 {2 a} \map \ln {\frac {x^2} {a^2 - x^2} } + C$
+\end{theorem}
+
+\begin{proof}
+With a view to expressing the [[Definition:Primitive|primitive]] in the form:
+:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
+let:
+{{begin-eqn}}
+{{eqn | l = u
+      | r = \tanh^{-1} \frac x a
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = \frac {\d u} {\d x}
+      | r = \frac a {a^2 - x^2}
+      | c = [[Derivative of Inverse Hyperbolic Tangent of x over a|Derivative of $\tanh^{-1} \dfrac x a$]]
+}}
+{{end-eqn}}
+and let:
+{{begin-eqn}}
+{{eqn | l = \frac {\d v} {\d x}
+      | r = \frac 1 {x^2}
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = v
+      | r = \frac {-1} x
+      | c = [[Primitive of Power]]
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = \int \frac {\tanh^{-1} \dfrac x a \rd x} {x^2}
+      | r = \paren {\tanh^{-1} \frac x a} \paren {\frac {-1} x} - \int \paren {\frac {-1} x} \paren {\frac a {a^2 - x^2} } \rd x + C
+      | c = [[Integration by Parts]]
+}}
+{{eqn | r = \frac {-\tanh^{-1} \dfrac x a} x + a \int \frac {\d x} {x \paren {a^2 - x^2} } + C
+      | c = simplifying
+}}
+{{eqn | r = \frac {-\tanh^{-1} \dfrac x a} x + a \paren {\frac 1 {2 a^2} \map \ln {\frac {x^2} {a^2 - x^2} } } + C
+      | c = [[Primitive of Reciprocal of x by a squared minus x squared|Primitive of $\dfrac 1 {x \paren {a^2 - x^2} }$]]
+}}
+{{eqn | r = \frac {-\tanh^{-1} \dfrac x a} x + \frac 1 {2 a} \map \ln {\frac {x^2} {a^2 - x^2} } + C
+      | c = simplifying
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of Inverse Hyperbolic Cotangent of x over a over x squared}
+Tags: Primitives involving Inverse Hyperbolic Cotangent Function
+
+\begin{theorem}
+:$\displaystyle \int \frac {\coth^{-1} \dfrac x a \rd x} {x^2} = \frac {-\coth^{-1} \dfrac x a} x + \frac 1 {2 a} \map \ln {\frac {x^2} {x^2 - a^2} } + C$
+\end{theorem}
+
+\begin{proof}
+With a view to expressing the [[Definition:Primitive|primitive]] in the form:
+:$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
+let:
+{{begin-eqn}}
+{{eqn | l = u
+      | r = \coth^{-1} \frac x a
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = \frac {\d u} {\d x}
+      | r = \frac {-a} {x^2 - a^2}
+      | c = [[Derivative of Inverse Hyperbolic Cotangent of x over a|Derivative of $\coth^{-1} \dfrac x a$]]
+}}
+{{end-eqn}}
+and let:
+{{begin-eqn}}
+{{eqn | l = \frac {\d v} {\d x}
+      | r = \frac 1 {x^2}
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = v
+      | r = \frac {-1} x
+      | c = [[Primitive of Power]]
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = \int \frac {\coth^{-1} \dfrac x a \rd x} {x^2}
+      | r = \paren {\coth^{-1} \frac x a} \paren {\frac {-1} x} - \int \paren {\frac {-1} x} \paren {\frac {-a} {x^2 - a^2} } \rd x + C
+      | c = [[Integration by Parts]]
+}}
+{{eqn | r = \frac {-\coth^{-1} \dfrac x a} x - a \int \frac {\d x} {x \paren {x^2 - a^2} } + C
+      | c = simplifying
+}}
+{{eqn | r = \frac {-\coth^{-1} \dfrac x a} x - a \paren {\frac 1 {2 a^2} \map \ln {\frac {x^2 - a^2} {x^2} } } + C
+      | c = [[Primitive of Reciprocal of x by x squared minus a squared|Primitive of $\dfrac 1 {x \paren {x^2 - a^2} }$]]
+}}
+{{eqn | r = \frac {-\coth^{-1} \dfrac x a} x - \frac 1 {2 a} \map \ln {\frac {x^2 - a^2} {x^2} } + C
+      | c = simplifying
+}}
+{{eqn | r = \frac {-\coth^{-1} \dfrac x a} x + \frac 1 {2 a} \map \ln {\frac {x^2} {x^2 - a^2} } + C
+      | c = [[Logarithm of Reciprocal]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of Power of x by Inverse Hyperbolic Sine of x over a}
+Tags: Primitives involving Inverse Hyperbolic Sine Function
+
+\begin{theorem}
+:$\displaystyle \int x^m \sinh^{-1} \frac x a \ \mathrm d x = \frac {x^{m + 1} } {m + 1} \sinh^{-1} \frac x a- \frac 1 {m + 1}  \int \frac {x^{m + 1} } {\sqrt {x^2 + a^2} } \ \mathrm d x + C$
+\end{theorem}
+
+\begin{proof}
+With a view to expressing the [[Definition:Primitive|primitive]] in the form:
+:$\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$
+let:
+{{begin-eqn}}
+{{eqn | l = u
+      | r = \sinh^{-1} \frac x a
+      | c = 
+}}
+{{eqn | ll= \implies
+      | l = \frac {\mathrm d u} {\mathrm d x}
+      | r = \frac 1 {\sqrt {x^2 + a^2} }
+      | c = [[Derivative of Inverse Hyperbolic Sine of x over a|Derivative of $\sinh^{-1} \dfrac x a$]]
+}}
+{{end-eqn}}
+and let:
+{{begin-eqn}}
+{{eqn | l = \frac {\mathrm d v} {\mathrm d x}
+      | r = x^m
+      | c = 
+}}
+{{eqn | ll= \implies
+      | l = v
+      | r = \frac {x^{m + 1} } {m + 1}
+      | c = [[Primitive of Power]]
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = \int \frac {\sinh^{-1} \dfrac x a \ \mathrm d x} {x^2}
+      | r = \left({\sinh^{-1} \frac x a}\right) \left({\frac {x^{m + 1} } {m + 1} }\right) - \int \left({\frac {x^{m + 1} } {m + 1} }\right) \left({\frac 1 {\sqrt {x^2 + a^2} } }\right) \ \mathrm d x + C
+      | c = [[Integration by Parts]]
+}}
+{{eqn | r = \frac {x^{m + 1} } {m + 1} \sinh^{-1} \frac x a - \frac 1 {m + 1}  \int \frac {x^{m + 1} } {\sqrt {x^2 + a^2} } \ \mathrm d x + C
+      | c = simplifying
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of Power of x by Inverse Hyperbolic Cotangent of x over a}
+Tags: Primitives involving Inverse Hyperbolic Cotangent Function
+
+\begin{theorem}
+:$\displaystyle \int x^m \coth^{-1} \frac x a \rd x = \frac {x^{m + 1} } {m + 1} \coth^{-1} \frac x a - \frac a {m + 1} \int \frac {x^{m + 1} } {a^2 - x^2} \rd x + C$
+\end{theorem}
+
+\begin{proof}
+With a view to expressing the [[Definition:Primitive|primitive]] in the form:
+:$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
+let:
+{{begin-eqn}}
+{{eqn | l = u
+      | r = \coth^{-1} \frac x a
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = \frac {\d u} {\d x}
+      | r = \frac {-a} {x^2 - a^2}
+      | c = [[Derivative of Inverse Hyperbolic Cotangent of x over a|Derivative of $\coth^{-1} \dfrac x a$]]
+}}
+{{end-eqn}}
+and let:
+{{begin-eqn}}
+{{eqn | l = \frac {\d v} {\d x}
+      | r = x^m
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = v
+      | r = \frac {x^{m + 1} } {m + 1}
+      | c = [[Primitive of Power]]
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = \int x^m \coth^{-1} \frac x a \rd x
+      | r = \paren {\coth^{-1} \frac x a} \paren {\frac {x^{m + 1} } {m + 1} } - \int \paren {\frac {x^{m + 1} } {m + 1} } \paren {\frac {-a} {x^2 - a^2} } \rd x + C
+      | c = [[Integration by Parts]]
+}}
+{{eqn | r = \frac {x^{m + 1} } {m + 1} \coth^{-1} \frac x a - \frac a {m + 1} \int \frac {x^{m + 1} } {a^2 - x^2} \rd x + C
+      | c = simplifying
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Primitive of Power of x by Inverse Hyperbolic Cosecant of x over a}
+Tags: Primitives involving Inverse Hyperbolic Cosecant Function
+
+\begin{theorem}
+:$\displaystyle \int x^m \operatorname{csch}^{-1} \frac x a \rd x = \begin{cases}
+\displaystyle \frac {x^{m + 1} } {m + 1} \operatorname{csch}^{-1} \frac x a + \frac 1 {m + 1} \int \frac {x^m} {\sqrt {x^2 + a^2} } \rd x + C & : x > 0 \\
+\displaystyle \frac {x^{m + 1} } {m + 1} \operatorname{csch}^{-1} \frac x a - \frac 1 {m + 1} \int \frac {x^m} {\sqrt {x^2 + a^2} } \rd x + C & : x < 0 \\
+\end{cases}$
+\end{theorem}
+
+\begin{proof}
+With a view to expressing the [[Definition:Primitive|primitive]] in the form:
+:$\displaystyle \int u \frac {\rd v} {\rd x} \rd x = u v - \int v \frac {\rd u} {\rd x} \rd x$
+let:
+{{begin-eqn}}
+{{eqn | l = u
+      | r = \operatorname{sech}^{-1} \frac x a
+      | c = 
+}}
+{{eqn | ll= \implies
+      | l = \frac {\rd u} {\rd x}
+      | r = \frac {-a} {\left\vert{x}\right\vert \sqrt{a^2 + x^2} }
+      | c = [[Derivative of Inverse Hyperbolic Cosecant of x over a|Derivative of $\operatorname{csch}^{-1} \dfrac x a$]]
+}}
+{{end-eqn}}
+and let:
+{{begin-eqn}}
+{{eqn | l = \frac {\rd v} {\rd x}
+      | r = x^m
+      | c = 
+}}
+{{eqn | ll= \implies
+      | l = v
+      | r = \frac {x^{m + 1} } {m + 1}
+      | c = [[Primitive of Power]]
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = \int \frac {\operatorname{csch}^{-1} \dfrac x a \rd x} {x^2}
+      | r = \left({\operatorname{csch}^{-1} \frac x a}\right) \left({\frac {x^{m + 1} } {m + 1} }\right) - \int \left({\frac {x^{m + 1} } {m + 1} }\right) \left({\frac {-a} {\left\vert{x}\right\vert \sqrt{a^2 + x^2} } }\right) \rd x + C
+      | c = [[Integration by Parts]]
+}}
+{{eqn | r = \begin{cases}
+\displaystyle \frac {x^{m + 1} } {m + 1} \operatorname{csch}^{-1} \frac x a + \frac 1 {m + 1} \int \frac {x^m} {\sqrt {x^2 + a^2} } \rd x + C & : x > 0 \\
+\displaystyle \frac {x^{m + 1} } {m + 1} \operatorname{csch}^{-1} \frac x a - \frac 1 {m + 1} \int \frac {x^m} {\sqrt {x^2 + a^2} } \rd x + C & : x < 0 \\
+\end{cases}
+      | c = Definition of [[Definition:Absolute Value|Absolute Value]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Hyperbolic Tangent Half-Angle Substitution}
+Tags: Integral Substitutions, Hyperbolic Tangent Function, Primitives involving Hyperbolic Sine Function, Primitives involving Hyperbolic Cosine Function, Hyperbolic Tangent Half-Angle Substitutions
+
+\begin{theorem}
+:$\displaystyle \int \map F {\sinh x, \cosh x} \rd x = 2 \int \map F {\frac {2 u} {1 - u^2}, \frac {1 + u^2} {1 - u^2} } \frac {\d u} {1 - u^2}$
+where $u = \tanh \dfrac x 2$.
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = u
+      | r = \tanh \dfrac x 2
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = x
+      | r = 2 \tanh^{-1} u
+      | c = {{Defof|Inverse Hyperbolic Tangent|subdef = Real|index = 1}}
+}}
+{{eqn | ll= \leadsto
+      | l = \dfrac {\d x} {\d u}
+      | r = \dfrac 2 {1 - u^2}
+      | c = [[Derivative of Inverse Hyperbolic Tangent]] and [[Derivative of Constant Multiple]]
+}}
+{{eqn | l = \sinh x
+      | r = \dfrac {2 u} {1 - u^2}
+      | c = [[Hyperbolic Tangent Half-Angle Substitution for Sine]]
+}}
+{{eqn | l = \cosh x
+      | r = \dfrac {1 + u^2} {1 - u^2}
+      | c = [[Hyperbolic Tangent Half-Angle Substitution for Cosine]]
+}}
+{{end-eqn}}
+The result follows from [[Integration by Substitution]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Reciprocal of One Plus Cosine}
+Tags: Trigonometric Identities, Cosine Function, Reciprocal of One Plus Cosine
+
+\begin{theorem}
+:$\dfrac 1 {1 + \cos x} = \dfrac 1 2 \sec^2 \dfrac x 2$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 1 + \cos x
+      | r = \cos 0 + \cos x
+      | c = [[Cosine of Zero is One]]
+}}
+{{eqn | r = 2 \map \cos {\dfrac {0 + x} 2} \map \cos {\dfrac {0 - x} 2}
+      | c = [[Prosthaphaeresis Formula for Cosine plus Cosine]]
+}}
+{{eqn | r = 2 \map \cos {\dfrac x 2} \map \cos {\dfrac {-x} 2}
+      | c = simplifying
+}}
+{{eqn | r = 2 \map \cos {\dfrac x 2} \map \cos {\dfrac x 2}
+      | c = [[Cosine Function is Even]]
+}}
+{{eqn | r = 2 \map {\cos^2} {\frac x 2}
+      | c = simplifying
+}}
+{{eqn | ll= \leadsto
+      | l = \frac 1 {1 + \cos x}
+      | r = \frac 1 2 \map {\sec^2} {\frac x 2}
+      | c = {{Defof|Secant Function}}
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \cos x
+      | r = 2 \cos^2 \frac x 2 - 1
+      | c = [[Double Angle Formulas/Cosine/Corollary 1|Cosine Double Angle Formula]]
+}}
+{{eqn | ll= \leadstoandfrom
+      | l = 1 + \cos x
+      | r = 2 \cos^2 \frac x 2
+      | c = adding $1$ to both sides
+}}
+{{eqn | ll= \leadstoandfrom
+      | l = \frac 1 {1 + \cos x}
+      | r = \frac 1 2 \frac 1 {\cos^2 \frac x 2}
+      | c = taking the [[Definition:Reciprocal|reciprocal]] of both sides
+}}
+{{eqn | r = \frac 1 2 \sec^2 \frac x 2
+      | c = {{Defof|Secant Function}}
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn	| l = \frac 1 {1 + \cos x} 
+	| r = \frac 1 {1 + \frac {1- \tan^2 \frac x 2} {1 + \tan^2 \frac x 2} }
+	| c = [[Tangent Half-Angle Substitution for Cosine]]
+}}
+{{eqn	| r = \frac {1 + \tan^2 \frac x 2} 2
+	| c = multiplying through $\frac {1 + \tan^2 \frac x 2} {1 + \tan^2 \frac x 2}$
+}}
+{{eqn	| r = \frac 1 2 \sec^2 \frac x 2
+	| c = [[Difference of Squares of Secant and Tangent]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Reciprocal of One Minus Cosine}
+Tags: Trigonometric Identities, Cosine Function, Reciprocal of One Minus Cosine
+
+\begin{theorem}
+:$\dfrac 1 {1 - \cos x} = \dfrac 1 2 \csc^2 \left({\dfrac x 2}\right)$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = 1 - \cos x
+      | r = \cos 0 - \cos x
+      | c = [[Cosine of Zero is One]]
+}}
+{{eqn | r = -2 \sin \left({\dfrac {0 + x} 2}\right) \sin \left({\dfrac {0 - x} 2}\right)
+      | c = [[Prosthaphaeresis Formula for Cosine minus Cosine]]
+}}
+{{eqn | r = -2 \sin \left({\dfrac x 2}\right) \sin \left({\dfrac {-x} 2}\right)
+      | c = simplifying
+}}
+{{eqn | r = 2 \sin \left({\dfrac x 2}\right) \sin \left({\dfrac x 2}\right)
+      | c = [[Sine Function is Odd]]
+}}
+{{eqn | r = 2 \sin^2 \left({\frac x 2}\right)
+      | c = simplifying
+}}
+{{eqn | ll= \implies
+      | l = \frac 1 {1 - \cos x}
+      | r = \frac 1 2 \csc^2 \left({\frac x 2}\right)
+      | c = {{Defof|Cosecant}}
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \cos x
+      | r = 1 - 2 \sin^2 \frac x 2
+      | c = [[Double Angle Formulas/Cosine/Corollary 2|Cosine Double Angle Formula]]
+}}
+{{eqn | ll= \iff
+      | l = 1 - \cos x
+      | r = 2 \sin^2 \frac x 2
+      | c = rearranging
+}}
+{{eqn | ll= \iff
+      | l = \frac 1 {1 - \cos x}
+      | r = \frac 1 2 \frac 1 {\sin^2 \frac x 2}
+      | c = taking the [[Definition:Reciprocal|reciprocal]] of both sides
+}}
+{{eqn | r = \frac 1 2 \csc^2 \frac x 2
+      | c = Definition of [[Definition:Cosecant|Cosecant]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn	| l = \frac 1 {1 - \cos x}
+	| r = \frac 1 {1 - \frac {1 - \map {\tan^2} {\frac x 2} } {1 + \map {\tan^2} {\frac x 2} } }
+	| c = [[Tangent Half-Angle Substitution for Cosine]]
+}}
+{{eqn	| r = \frac {1 + \map {\tan^2} {\frac x 2} } {1 + \map {\tan^2} {\frac x 2} - 1 + \map {\tan^2} {\frac x 2} }
+	| c = multiplying through $\frac {1 + \map {\tan^2} {\frac x 2} } {1 + \map {\tan^2} {\frac x 2} }$
+}}
+{{eqn	| r = \frac {\map {\sec^2} {\frac x 2} } {2 \map {\tan^2} {\frac x 2} }
+	| c = [[Difference of Squares of Secant and Tangent]]
+}}
+{{eqn	| r = \frac 1 2 \cdot \frac 1 {\map {\cos^2} {\frac x 2} } \cdot \frac {\map {\cos^2} {\frac x 2} } {\map {\sin^2} {\frac x 2} }
+	| c = {{Defof|Secant Function}}, {{Defof|Tangent}}
+}}
+{{eqn	| r = \frac 1 {2 \map {\sin^2} {\frac x 2} }
+}}
+{{eqn	| r = \frac 1 2 \map {\csc^2} {\frac x 2}
+	| c = {{Defof|Cosecant}}
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Irrationality of Logarithm}
+Tags: Number Theory, Irrationality Proofs, Logarithms
+
+\begin{theorem}
+Let $a, b \in \N_{>0}$ such that both $\nexists m, n \in \N_{>0}: a^m = b^n$.
+Then $\log_b a$ is [[Definition:Irrational Number|irrational]].
+\end{theorem}
+
+\begin{proof}
+{{AimForCont}} $\log_b a$ is [[Definition:Rational Number|rational]].
+Then:
+:$\exists p, q \in \N_{>0} : \log_b a = \dfrac p q$
+where $p \perp q$.
+Then:
+{{begin-eqn}}
+{{eqn | l = \log_b a
+      | r = \dfrac p q
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = b^{\frac p q}
+      | r = a
+      | c = {{Defof|Real General Logarithm}}
+}}
+{{eqn | ll= \leadsto
+      | l = \sqrt [q] {\paren {b^p} }
+      | r = a
+      | c = {{Defof|Rational Power}}
+}}
+{{eqn | ll= \leadsto
+      | l = b^p
+      | r = a^q
+      | c = {{Defof|Root (Analysis)|Root}}
+}}
+{{end-eqn}}
+which [[Definition:Contradiction|contradicts]] the initial assumption:
+:$\nexists m, n \in \N: a^m = b^n$
+Hence the result by [[Proof by Contradiction]].
+{{qed}}
+[[Category:Number Theory]]
+[[Category:Irrationality Proofs]]
+[[Category:Logarithms]]
+qgxchhi9q3h3ade7m4iw14nq2jb189f
+\end{proof}<|endoftext|>
+\section{Tangent of 22.5 Degrees}
+Tags: Tangent Function
+
+\begin{theorem}
+:$\tan 22.5^\circ = \tan \dfrac \pi 8 = \sqrt 2 - 1$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \tan 22.5^\circ
+      | r = \tan \frac {45^\circ} 2
+      | c = 
+}}
+{{eqn | r = \frac {1 - \cos 45^\circ} {\sin 45^\circ}
+      | c = [[Half Angle Formulas/Tangent/Corollary 2]]
+}}
+{{eqn | r = \frac {1 - \frac {\sqrt 2} 2} {\frac {\sqrt 2} 2}
+      | c = [[Cosine of 45 Degrees]] and [[Sine of 45 Degrees]]
+}}
+{{eqn | r = \sqrt 2 - 1
+      | c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $\sqrt 2$
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Tangent Function]]
+1m3wwb0zdcdp5uazkbqzc1k5uga9wd2
+\end{proof}<|endoftext|>
+\section{Tangent of 67.5 Degrees}
+Tags: Tangent Function
+
+\begin{theorem}
+:$\tan 67.5 \degrees = \tan \dfrac {3 \pi} 8 = \sqrt 2 + 1$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \tan 67.5 \degrees
+      | r = \map \tan {45 \degrees + 22.5 \degrees}
+      | c = 
+}}
+{{eqn | r = \frac {\tan 45 \degrees + \tan 22.5 \degrees} {1 - \tan 45 \degrees \tan 22.5 \degrees}
+      | c = [[Tangent of Sum]]
+}}
+{{eqn | r = \frac {1 + \paren {\sqrt 2 - 1} } {1 - 1 \times \paren {\sqrt 2 - 1} }
+      | c = [[Tangent of 45 Degrees|Tangent of $45 \degrees$]] and [[Tangent of 22.5 Degrees|Tangent of $22.5 \degrees$]]
+}}
+{{eqn | r = \frac {\sqrt 2} {2 - \sqrt 2}
+      | c = simplifying
+}}
+{{eqn | r = \frac {\sqrt 2 \paren {2 + \sqrt 2} } {\paren {2 - \sqrt 2} \paren {2 + \sqrt 2} }
+      | c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $2 + \sqrt 2$
+}}
+{{eqn | r = \frac {2 \sqrt 2 + 2} {4 - 2}
+      | c = [[Difference of Two Squares]]
+}}
+{{eqn | r = \sqrt 2 + 1
+      | c = simplifying
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Tangent Function]]
+hul6jsgdvh6kjjq1uwbjqvmiu04qpvy
+\end{proof}<|endoftext|>
+\section{Sine of x plus Cosine of x/Cosine Form}
+Tags: Sine Function, Cosine Function
+
+\begin{theorem}
+:$\sin x + \cos x = \sqrt 2 \, \map \cos {x - \dfrac \pi 4}$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \sin x + \cos x
+      | r = \sin x + \map \sin {\frac \pi 2 - x}
+      | c = [[Sine of Complement equals Cosine]]
+}}
+{{eqn | r = 2 \, \map \sin {\frac {x + \paren {\frac \pi 2 - x} } 2} \map \cos {\frac {x - \paren {\frac \pi 2 - x} } 2}
+      | c = [[Prosthaphaeresis Formula for Sine plus Sine]]
+}}
+{{eqn | r = 2 \sin \frac \pi 4 \, \map \cos {x - \frac \pi 4}
+      | c = simplifying
+}}
+{{eqn | r = \sqrt 2 \, \map \cos {x - \frac \pi 4}
+      | c = [[Sine of 45 Degrees|Sine of $\dfrac \pi 4$]]
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Sine Function]]
+[[Category:Cosine Function]]
+ajw6vg0ofm5gr79irlb0v9tmr1erk5t
+\end{proof}<|endoftext|>
+\section{Sine of x minus Cosine of x/Sine Form}
+Tags: Sine Function, Cosine Function
+
+\begin{theorem}
+:$\sin x - \cos x = \sqrt 2 \sin \left({x - \dfrac \pi 4}\right)$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \sin x - \cos x
+      | r = \sin x - \sin \left({\frac \pi 2 - x}\right)
+      | c = [[Sine of Complement equals Cosine]]
+}}
+{{eqn | r = 2 \cos \left({\frac {x + \left({\frac \pi 2 - x}\right)} 2}\right) \sin \left({\frac {x - \left({\frac \pi 2 - x}\right)} 2}\right)
+      | c = [[Prosthaphaeresis Formula for Sine minus Sine]]
+}}
+{{eqn | r = 2 \cos \frac \pi 4 \sin \left({x - \frac \pi 4}\right)
+      | c = simplifying
+}}
+{{eqn | r = \sqrt 2 \sin \left({x - \frac \pi 4}\right)
+      | c = [[Cosine of 45 Degrees|Cosine of $\dfrac \pi 4$]]
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Sine Function]]
+[[Category:Cosine Function]]
+mhg93r8g7ltbh2e0t9nwjzdio0bfyfm
+\end{proof}<|endoftext|>
+\section{Sine of x minus Cosine of x}
+Tags: Sine Function, Cosine Function
+
+\begin{theorem}
+=== [[Sine of x minus Cosine of x/Sine Form|Sine Form]] ===
+{{:Sine of x minus Cosine of x/Sine Form}}
+=== [[Sine of x minus Cosine of x/Cosine Form|Cosine Form]] ===
+{{:Sine of x minus Cosine of x/Cosine Form}}
+\end{theorem}<|endoftext|>
+\section{Sine of x minus Cosine of x/Cosine Form}
+Tags: Sine Function, Cosine Function
+
+\begin{theorem}
+:$\sin x - \cos x = \sqrt 2 \, \map \cos {x - \dfrac {3 \pi} 4}$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \sin x - \cos x
+      | r = \sqrt 2 \, \map \sin {x - \dfrac \pi 4}
+      | c = [[Sine of x minus Cosine of x/Sine Form|Sine of x minus Cosine of x: Sine Form]]
+}}
+{{eqn | r = \sqrt 2 \, \map \cos {\frac \pi 2 - \paren {x - \dfrac \pi 4} }
+      | c = [[Cosine of Complement equals Sine]]
+}}
+{{eqn | r = \sqrt 2 \, \map \cos {\frac \pi 2 - x + \dfrac \pi 4}
+      | c = simplifying
+}}
+{{eqn | r = \sqrt 2 \, \map \cos {\frac {3 \pi} 4 - x}
+      | c = simplifying
+}}
+{{eqn | r = \sqrt 2 \, \map \cos {x - \frac {3 \pi} 4}
+      | c = [[Cosine Function is Even]]
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Sine Function]]
+[[Category:Cosine Function]]
+4w1mna76rlqbh5e3l9cnpda8q6g5jgw
+\end{proof}<|endoftext|>
+\section{Multiple of Sine plus Multiple of Cosine}
+Tags: Sine Function, Cosine Function
+
+\begin{theorem}
+=== [[Multiple of Sine plus Multiple of Cosine/Cosine Form|Cosine Form]] ===
+{{:Multiple of Sine plus Multiple of Cosine/Cosine Form}}
+=== [[Multiple of Sine plus Multiple of Cosine/Sine Form|Sine Form]] ===
+{{:Multiple of Sine plus Multiple of Cosine/Sine Form}}
+\end{theorem}<|endoftext|>
+\section{Sum Formulas for Sine and Cosine}
+Tags: Trigonometric Identities, Hyperbolic Functions
+
+\begin{theorem}
+=== [[Sine of x plus Cosine of x/Sine Form|Sine of x plus Cosine of x: Sine Form]] ===
+{{:Sine of x plus Cosine of x/Sine Form}}
+=== [[Sine of x plus Cosine of x/Cosine Form|Sine of x plus Cosine of x: Cosine Form]] ===
+{{:Sine of x plus Cosine of x/Cosine Form}}
+=== [[Sine of x minus Cosine of x/Sine Form|Sine of x minus Cosine of x: Sine Form]] ===
+{{:Sine of x minus Cosine of x/Sine Form}}
+=== [[Sine of x minus Cosine of x/Cosine Form|Sine of x minus Cosine of x: Cosine Form]] ===
+{{:Sine of x minus Cosine of x/Cosine Form}}
+=== [[Cosine of x minus Sine of x/Sine Form|Cosine of x minus Sine of x: Sine Form]] ===
+{{:Cosine of x minus Sine of x/Sine Form}}
+=== [[Cosine of x minus Sine of x/Cosine Form|Cosine of x minus Sine of x: Cosine Form]] ===
+{{:Cosine of x minus Sine of x/Cosine Form}}
+== [[Multiple of Sine plus Multiple of Cosine]] ==
+{{:Multiple of Sine plus Multiple of Cosine}}
+\end{theorem}<|endoftext|>
+\section{Multiple of Sine plus Multiple of Cosine/Cosine Form}
+Tags: Sine Function, Cosine Function
+
+\begin{theorem}
+:$p \sin x + q \cos x = \sqrt {p^2 + q^2} \map \cos {x + \arctan \dfrac {-p} q}$
+\end{theorem}
+
+\begin{proof}
+Let it be assumed that $p \sin x + q \cos x$ can be expressed in the form $M \map \cos {x + \phi}$.
+Then:
+{{begin-eqn}}
+{{eqn | l = p \sin x + q \cos x
+      | r = M \cos \map \cos {x + \phi}
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = \frac p M \sin x + \frac q M \cos x
+      | r = \map \cos {x + \phi}
+      | c = 
+}}
+{{eqn | r = \cos x \cos \phi - \sin x \sin \phi
+      | c = [[Cosine of Sum]]
+}}
+{{eqn | ll= \leadsto
+      | l = \sin x \sin \phi
+      | r = \frac {-p} M \sin x
+      | c = equating terms in $\sin x$
+}}
+{{eqn | l = \cos x \cos \phi
+      | r = \frac q M \cos x
+      | c =  and $\cos x$
+}}
+{{eqn | n = 1
+      | ll= \leadsto
+      | l = \sin \phi
+      | r = \frac {-p} M
+      | c = 
+}}
+{{eqn | n = 2
+      | l = \cos \phi
+      | r = \frac q M
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = \sin^2 \phi + \cos^2 \phi
+      | r = \paren {\frac {-p} M}^2 + \paren {\frac q M}^2
+      | c = squaring and adding $(1)$ and $(2)$
+}}
+{{eqn | ll= \leadsto
+      | l = 1
+      | r = \frac {p^2 + q^2} {M^2}
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | r = M
+      | l = \sqrt {p^2 + q^2}
+      | c = 
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = \tan \phi
+      | r = \frac {\sin \phi} {\cos \phi}
+      | c = [[Tangent is Sine divided by Cosine]]
+}}
+{{eqn | r = \frac {-p / M} {q / M}
+      | c = from $(1)$ and $(2)$
+}}
+{{eqn | r = \frac {-p} q
+      | c = simplifying
+}}
+{{end-eqn}}
+Hence the result.
+{{qed}}
+[[Category:Sine Function]]
+[[Category:Cosine Function]]
+jr2opcgzj2mrev53xztoxdexi2w2o40
+\end{proof}<|endoftext|>
+\section{Multiple of Sine plus Multiple of Cosine/Sine Form}
+Tags: Sine Function, Cosine Function
+
+\begin{theorem}
+:$p \sin x + q \cos x = \sqrt {p^2 + q^2} \sin \left({x + \arctan \dfrac q p}\right)$
+\end{theorem}
+
+\begin{proof}
+Let it be assumed that $p \sin x + q \cos x$ can be expressed in the form $M \sin \left({x + \phi}\right)$.
+Then:
+{{begin-eqn}}
+{{eqn | l = p \sin x + q \cos x
+      | r = M \sin \left({x + \phi}\right)
+      | c = 
+}}
+{{eqn | ll= \implies
+      | l = \frac p M \sin x + \frac q M \cos x
+      | r = \sin \left({x + \phi}\right)
+      | c = 
+}}
+{{eqn | r = \sin x \cos \phi + \cos x \sin \phi
+      | c = [[Sine of Sum]]
+}}
+{{eqn | ll= \implies
+      | l = \sin x \cos \phi
+      | r = \frac p M \sin x
+      | c = equating terms in $\sin x$
+}}
+{{eqn | l = \cos x \sin \phi
+      | r = \frac q M \cos x
+      | c =  and $\cos x$
+}}
+{{eqn | n = 1
+      | ll= \implies
+      | l = \sin \phi
+      | r = \frac q M
+      | c = 
+}}
+{{eqn | n = 2
+      | l = \cos \phi
+      | r = \frac p M
+      | c = 
+}}
+{{eqn | ll= \implies
+      | l = \sin^2 \phi + \cos^2 \phi
+      | r = \left({\frac q M}\right)^2 + \left({\frac p M}\right)^2
+      | c = squaring and adding $(1)$ and $(2)$
+}}
+{{eqn | ll= \implies
+      | l = 1
+      | r = \frac {p^2 + q^2} {M^2}
+      | c = 
+}}
+{{eqn | ll= \implies
+      | r = M
+      | l = \sqrt {p^2 + q^2}
+      | c = 
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = \tan \phi
+      | r = \frac {\sin \phi} {\cos \phi}
+      | c = [[Tangent is Sine divided by Cosine]]
+}}
+{{eqn | r = \frac {q / M} {p / M}
+      | c = from $(1)$ and $(2)$
+}}
+{{eqn | r = \frac q p
+      | c = simplifying
+}}
+{{end-eqn}}
+Hence the result.
+{{qed}}
+[[Category:Sine Function]]
+[[Category:Cosine Function]]
+oc56d623463q7jxezd0o7ngan8b9zd8
+\end{proof}<|endoftext|>
+\section{Primitive of Cosine of a x over Sine of a x plus phi}
+Tags: Primitives involving Sine Function, Primitives involving Cosine Function
+
+\begin{theorem}
+:$\displaystyle \int \frac {\cos a x \rd x} {\map \sin {a x + \phi} } = \frac {\ln \size {\map \sin {a x + \phi} } } {a \cos \phi} + \tan \phi \int \frac {\sin a x \rd x} {\map \sin {a x + \phi} } + C$
+\end{theorem}
+
+\begin{proof}
+First note that:
+{{begin-eqn}}
+{{eqn | n = 1
+      | l = \map {\frac \d {\d x} } {\map \sin {a x + \phi} }
+      | r = a \, \map \cos {a x + \phi}
+      | c = [[Derivative of Sine of a x|Derivative of $\sin a x$]] etc.
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = \int \frac {\cos a x \rd x} {\map \sin {a x + \phi} }
+      | r = \int \frac {\cos a x \cos \phi \rd x} {\cos \phi \, \map \sin {a x + \phi} }
+      | c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $\cos \phi$
+}}
+{{eqn | r = \int \frac {\paren {\cos a x \cos \phi - \sin a x \sin \phi + \sin a x \sin \phi} \rd x} {\cos \phi \, \map \sin {a x + \phi} }
+      | c = 
+}}
+{{eqn | r = \int \frac {\paren {\map \cos {a x + \phi} + \sin a x \sin \phi} \rd x} {\cos \phi \, \map \sin {a x + \phi} }
+      | c = [[Cosine of Sum]]
+}}
+{{eqn | r = \frac 1 {\cos \phi} \int \frac {\map \cos {a x + \phi} \rd x} {\map \sin {a x + \phi} } + \frac {\sin \phi} {\cos \phi} \int \frac {\sin a x \rd x} {\map \sin {a x + \phi} }
+      | c = [[Linear Combination of Integrals]]
+}}
+{{eqn | r = \frac 1 {\cos \phi} \ln \size {\map \sin {a x + \phi} } + \frac {\sin \phi} {\cos \phi} \int \frac {\sin a x \rd x} {\map \sin {a x + \phi} }
+      | c = [[Primitive of Function under its Derivative]] and $(1)$
+}}
+{{eqn | r = \frac {\ln \size {\map \sin {a x + \phi} } } {a \cos \phi} + \tan \phi \int \frac {\sin a x \rd x} {\map \sin {a x + \phi} } + C
+      | c = [[Tangent is Sine divided by Cosine]]
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Primitives involving Sine Function]]
+[[Category:Primitives involving Cosine Function]]
+h6zl6lhm66diinvu2vcr0bv7k58ai5z
+\end{proof}<|endoftext|>
+\section{Primitive of Sine of a x over Sine of a x plus phi}
+Tags: Primitives involving Sine Function, Primitives involving Cosine Function
+
+\begin{theorem}
+:$\displaystyle \int \frac {\sin a x \ \mathrm d x} {\sin \left({a x + \phi}\right)} = \frac x {\cos \phi} - \tan \phi \int \frac {\cos a x \ \mathrm d x} {\sin \left({a x + \phi}\right)} + C$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \int \frac {\sin a x \ \mathrm d x} {\sin \left({a x + \phi}\right)}
+      | r = \frac 1 {\cos \phi} \int \frac {\sin a x \cos \phi \ \mathrm d x} {\sin \left({a x + \phi}\right)}
+      | c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $\cos \phi$
+}}
+{{eqn | r = \frac 1 {\cos \phi} \int \frac {\left({\sin a x \cos \phi + \cos a x \sin \phi - \cos a x \sin \phi}\right) \ \mathrm d x} {\sin \left({a x + \phi}\right)}
+      | c = 
+}}
+{{eqn | r = \frac 1 {\cos \phi} \int \frac {\left({\sin \left({a x + \phi}\right) - \cos a x \sin \phi}\right) \ \mathrm d x} {\sin \left({a x + \phi}\right)}
+      | c = [[Sine of Sum]]
+}}
+{{eqn | r = \frac 1 {\cos \phi} \int \frac {\left({\sin \left({a x + \phi}\right) - \cos a x \sin \phi}\right) \ \mathrm d x} {\sin \left({a x + \phi}\right)}
+      | c = [[Sine of Sum]]
+}}
+{{eqn | r = \frac 1 {\cos \phi} \int \mathrm d x  - \frac {\sin \phi} {\cos \phi} \int \frac {\cos a x \ \mathrm d x} {\sin \left({a x + \phi}\right)}
+      | c = [[Linear Combination of Integrals]]
+}}
+{{eqn | r = \frac x {\cos \phi} - \tan \phi \int \frac {\cos a x \ \mathrm d x} {\sin \left({a x + \phi}\right)} + C
+      | c = [[Tangent is Sine divided by Cosine]]
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Primitives involving Sine Function]]
+[[Category:Primitives involving Cosine Function]]
+8u5q1cmoonorym0tk2lhyqd4ysmepto
+\end{proof}<|endoftext|>
+\section{Reciprocal of One Plus Hyperbolic Cosine}
+Tags: Hyperbolic Cosine Function
+
+\begin{theorem}
+:$\dfrac 1 {1 + \cosh x} = \dfrac 1 2 \operatorname{sech}^2 \dfrac x 2$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \cosh x
+      | r = 2 \cosh^2 \frac x 2 - 1
+      | c = [[Double Angle Formulas/Hyperbolic Cosine/Corollary 1|Hyperbolic Cosine Double Angle Formula]]
+}}
+{{eqn | ll= \iff
+      | l = 1 + \cosh x
+      | r = 2 \cosh^2 \frac x 2
+      | c = adding $1$ to both sides
+}}
+{{eqn | ll= \iff
+      | l = \frac 1 {1 + \cosh x}
+      | r = \frac 1 2 \frac 1 {\cosh^2 \frac x 2}
+      | c = taking the [[Definition:Reciprocal|reciprocal]] of both sides
+}}
+{{eqn | r = \frac 1 2 \operatorname{sech}^2 \frac x 2
+      | c = Definition of [[Definition:Hyperbolic Secant/Definition 2|Hyperbolic Secant]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Reciprocal of Hyperbolic Cosine Minus One}
+Tags: Hyperbolic Cosine Function
+
+\begin{theorem}
+:$\dfrac 1 {\cosh x - 1} = \dfrac 1 2 \csch^2 \dfrac x 2$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \cosh x
+      | r = 1 + 2 \sinh^2 \frac x 2
+      | c = [[Double Angle Formulas/Hyperbolic Cosine/Corollary 2|Hyperbolic Cosine Double Angle Formula]]
+}}
+{{eqn | ll= \leadstoandfrom
+      | l = \cosh x - 1
+      | r = 2 \sinh^2 \frac x 2
+      | c = subtracting $1$ from both sides
+}}
+{{eqn | ll= \leadstoandfrom
+      | l = \frac 1 {\cosh x - 1}
+      | r = \frac 1 2 \frac 1 {\sinh^2 \frac x 2}
+      | c = taking the [[Definition:Reciprocal|reciprocal]] of both sides
+}}
+{{eqn | r = \frac 1 2 \csch^2 \frac x 2
+      | c = {{Defof|Hyperbolic Cosecant|index = 2}}
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Steiner-Lehmus Theorem}
+Tags: Triangles, Steiner-Lehmus Theorem
+
+\begin{theorem}
+Let $ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
+Denote the lengths of the [[Definition:Angle Bisector|angle bisectors]] through the [[Definition:Vertex of Polygon|vertices]] $A$ and $B$ by $\omega_\alpha$ and $\omega_\beta$.
+Let $\omega_\alpha = \omega_\beta$.
+Then $ABC$ is an [[Definition:Isosceles Triangle|isosceles triangle]].
+\end{theorem}
+
+\begin{proof}
+Let $a$, $b$, and $c$ are the [[Definition:Opposite (in Triangle)|sides opposite]] $A$, $B$ and $C$ respectively.
+By [[Length of Angle Bisector]], $\omega_\alpha, \omega_\beta$ are given by:
+:$\omega_\alpha^2 = \dfrac {b c} {\paren {b + c}^2} \paren {\paren {b + c}^2 - a^2}$
+:$\omega_\beta^2 = \dfrac {b a} {\paren {b + a}^2} \paren {\paren {b + a}^2 - c^2}$
+Equating the two equations, 
+{{begin-eqn}}
+{{eqn | l = \dfrac {b c} {\paren {b + c}^2} \paren {\paren {b + c}^2 - a^2}
+      | r = \dfrac {b a} {\paren {b + a}^2} \paren {\paren {b + a}^2 - c^2}
+}}
+{{eqn | ll = \leadsto
+      | l = b c \paren {b + a}^2 \paren {\paren {b + c}^2 - a^2}
+      | r = b a \paren {b + c}^2 \paren {\paren {b + a}^2 - c^2}
+}}
+{{eqn | ll = \leadsto
+      | l = b c \paren {b + a}^2 \paren {b + c + a} \paren {b + c - a}
+      | r = b a \paren {b + c}^2 \paren {b + a + c} \paren {b + a - c}
+}}
+{{eqn | ll = \leadsto
+      | l = c \paren {b + a}^2 \paren {b + c - a}
+      | r = a \paren {b + c}^2 \paren {b + a - c} \quad \quad a + b + c > 0 \text { as each } a, b, c > 0
+}}
+{{eqn | ll = \leadsto
+      | l = a^2 b c + a^2 c^2 - a^3 c + 2 a b^2 c + 2 a b c^2 - 2 a^2 b c + b^3 c + b^2 c^2 - a b^2 c
+      | r = a b^3 + a^2 b^2 - a b^2 c + 2 a b^2 c + 2 a^2 b c - 2 a b c^2 + a b c^2 + a^2 c^2 - a c^3
+}}
+{{eqn | ll = \leadsto
+      | l = - a^3 c + 2 a b c^2 - a^2 b c + b^3 c + b^2 c^2
+      | r = a b^3 + a^2 b^2 + 2 a^2 b c - a b c^2 - a c^3
+}}
+{{eqn | ll = \leadsto
+      | l = a c^3 - a^3 c + 3 a b c^2 - 3 a^2 b c + b^3 c - a b^3 + b^2 c^2 - a^2 b^2 
+      | r = 0
+}}
+{{eqn | ll = \leadsto
+      | l = \paren {c - a} \paren {a c \paren {c + a} + 3 a b c + b^3 + b^2 \paren {c + a} }
+      | r = 0
+}}
+{{eqn | ll = \leadsto
+      | l = c - a
+      | r = 0 \quad \quad a c \paren {c + a} + 3 a b c + b^3 + b^2 \paren {c + a} > 0 \text { as each } a, b, c > 0
+}}
+{{eqn | ll = \leadsto
+      | l = c
+      | r = a
+}}
+{{end-eqn}}
+Therefore $ABC$ is an [[Definition:Isosceles Triangle|isosceles triangle]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Triangle Right-Angle-Hypotenuse-Side Equality}
+Tags: Right Triangles
+
+\begin{theorem}
+If two [[Definition:Triangle (Geometry)|triangles]] have:
+: one [[Definition:Right Angle|right angle]] each
+: the sides opposite to the right angle equal
+: another two respective sides equal
+they will also have:
+: their third sides equal
+: the remaining two angles equal to their respective remaining angles.
+\end{theorem}
+
+\begin{proof}
+Let $\triangle ABC$ and $\triangle DEF$ be two [[Definition:Triangle (Geometry)|triangles]] having sides $AB = DE$ and $AC = DF$, and with $\angle ABC = \angle DEF = 90^\circ$.
+By [[Pythagoras' Theorem]]:
+: $BC = \sqrt {AB^2 + AC^2}$
+and:
+: $EF = \sqrt {DE^2 + DF^2}$
+:$\therefore BC = \sqrt {AB^2 + AC^2} = \sqrt {DE^2 + DF^2} = EF$
+The part that the remaining two angles are equal to their respective remaining angles follows from [[Triangle Side-Side-Side Equality]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Internal Angles of Regular Polygon}
+Tags: Regular Polygons
+
+\begin{theorem}
+The size $A$ of each [[Definition:Internal Angle|internal angle]] of a [[Definition:Regular Polygon|regular $n$-gon]] is given by:
+:$A = \dfrac {\paren {n - 2} 180 \degrees} n$
+\end{theorem}
+
+\begin{proof}
+From [[Sum of Internal Angles of Polygon]], we have that the sum $S$ of all [[Definition:Internal Angle|internal angles]] of a [[Definition:Polygon|$n$-gon]] is:
+:$S = \paren {n - 2} 180 \degrees$
+From the definition of a [[Definition:Regular Polygon|regular polygon]], all the [[Definition:Internal Angle|internal angles]] of a [[Definition:Regular Polygon|regular polygon]] are equal.
+Therefore, the size $A$ of each [[Definition:Internal Angle|internal angle]] of a [[Definition:Regular Polygon|regular polygon]] with $n$ [[Definition:Side of Polygon|sides]] is:
+:$A = \dfrac {\paren {n - 2} 180 \degrees} n$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Sum of Internal Angles of Polygon}
+Tags: Polygons
+
+\begin{theorem}
+The sum $S$ of all [[Definition:Internal Angle|internal angles]] of a [[Definition:Polygon|polygon]] with $n$ sides is given by the formula $S = \paren {n - 2} 180 \degrees$.
+\end{theorem}
+
+\begin{proof}
+For [[Definition:Convex Polygon|convex polygons]], name a [[Definition:Vertex of Polygon|vertex]] as $A_1$, go [[Definition:Clockwise|clockwise]] and name the [[Definition:Vertex of Polygon|vertices]] as $A_2, A_3, \ldots, A_n$.
+By joining $A_1$ to every [[Definition:Vertex of Polygon|vertex]] except $A_2$ and $A_n$, one can form $\paren {n - 2}$ [[Definition:Triangle (Geometry)|triangles]].
+From [[Sum of Angles of Triangle equals Two Right Angles]], the sum of the [[Definition:Internal Angle|internal angles]] of a  [[Definition:Triangle (Geometry)|triangle]] is $180 \degrees$.
+Therefore, the sum of [[Definition:Internal Angle|internal angles]] of a [[Definition:Polygon|polygon]] with $n$ sides is $\paren {n - 2} 180 \degrees$.
+{{qed}}
+{{proof wanted|concave polygons?}}
+\end{proof}<|endoftext|>
+\section{Five Platonic Solids}
+Tags: Platonic Solids, 5, Five Platonic Solids
+
+\begin{theorem}
+There exist exactly five [[Definition:Platonic Solid|platonic solids]]:
+:$\paren 1: \quad$ the [[Definition:Regular Tetrahedron|regular tetrahedron]]
+:$\paren 2: \quad$ the [[Definition:Cube (Geometry)|cube]]
+:$\paren 3: \quad$ the [[Definition:Regular Octahedron|regular octahedron]]
+:$\paren 4: \quad$ the [[Definition:Regular Dodecahedron|regular dodecahedron]]
+:$\paren 5: \quad$ the [[Definition:Regular Icosahedron|regular icosahedron]].
+{{:Euclid:Proposition/XIII/18/Endnote}}
+\end{theorem}
+
+\begin{proof}
+A [[Definition:Solid Angle|solid angle]] cannot be constructed from only two [[Definition:Plane|planes]].
+Therefore at least three [[Definition:Face of Polyhedron|faces]] need to come together to form a [[Definition:Vertex of Polyhedron|vertex]].
+Let $P$ be a [[Definition:Platonic Solid|platonic solid]].
+Let the [[Definition:Polygon|polygon]] which forms each [[Definition:Face of Polyhedron|face]] of $P$ be a [[Definition:Equilateral Triangle|equilateral triangles]].
+We have that:
+:each [[Definition:Vertex of Polyhedron|vertex]] of a [[Definition:Regular Tetrahedron|regular tetrahedron]] is composed of $3$ [[Definition:Equilateral Triangle|equilateral triangles]]
+:each [[Definition:Vertex of Polyhedron|vertex]] of a [[Definition:Regular Octahedron|regular octahedron]] is composed of $4$ [[Definition:Equilateral Triangle|equilateral triangles]]
+:each [[Definition:Vertex of Polyhedron|vertex]] of a [[Definition:Regular Icosahedron|regular icosahedron]] is composed of $5$ [[Definition:Equilateral Triangle|equilateral triangles]].
+$6$ [[Definition:Equilateral Triangle|equilateral triangles]], placed together at a [[Definition:Vertex of Polyhedron|vertex]], form $4$ [[Definition:Right Angle|right angles]].
+From {{EuclidPropLink|book = XI|prop = 21|title = Solid Angle contained by Plane Angles is Less than Four Right Angles}}:
+:a [[Definition:Solid Angle|solid angle]] is [[Definition:Containment of Solid Angle|contained]] by [[Definition:Plane Angle|plane angles]] which total less than $4$ [[Definition:Right Angle|right angles]].
+Thus it is not possible to form $P$ such that its [[Definition:Vertex of Polyhedron|vertices]] are formed by $6$ [[Definition:Equilateral Triangle|equilateral triangles]].
+For the same reason, it is not possible to form $P$ such that its [[Definition:Vertex of Polyhedron|vertices]] are formed by more than $6$ [[Definition:Equilateral Triangle|equilateral triangles]].
+Hence there are only $3$ possible [[Definition:Platonic Solid|platonic solids]] whose [[Definition:Face of Polyhedron|faces]] are [[Definition:Equilateral Triangle|equilateral triangles]].
+We have that each [[Definition:Vertex of Polyhedron|vertex]] of a [[Definition:Cube|cube]] is composed of $3$ [[Definition:Square (Geometry)|squares]].
+$4$ [[Definition:Square (Geometry)|squares]], placed together at a [[Definition:Vertex of Polyhedron|vertex]], form $4$ [[Definition:Right Angle|right angles]].
+Thus, again from {{EuclidPropLink|book = XI|prop = 21|title = Solid Angle contained by Plane Angles is Less than Four Right Angles}}:
+:it is not possible to form $P$ such that its [[Definition:Vertex of Polyhedron|vertices]] are formed by $4$ [[Definition:Square (Geometry)|squares]].
+For the same reason, it is not possible to form $P$ such that its [[Definition:Vertex of Polyhedron|vertices]] are formed by more than $4$ [[Definition:Square (Geometry)|squares]].
+Hence there is only $1$ possible [[Definition:Platonic Solid|platonic solid]] whose [[Definition:Face of Polyhedron|faces]] are [[Definition:Square (Geometry)|squares]].
+We have that each [[Definition:Vertex of Polyhedron|vertex]] of a [[Definition:Regular Dodecahedron|regular dodecahedron]] is composed of $3$ [[Definition:Regular Pentagon|regular pentagons]].
+From {{EuclidPropLink|book = XIII|prop = 18|qual=Lemma|title = Comparison of Sides of Five Platonic Figures}}:
+:the [[Definition:Vertex of Polygon|vertices]] of a [[Definition:Regular Pentagon|regular pentagon]] equal $1 \dfrac 1 5$ [[Definition:Right Angle|right angles]].
+$4$ [[Definition:Regular Pentagon|regular pentagons]], placed together at a [[Definition:Vertex of Polyhedron|vertex]], form $4 \dfrac 4 5$ [[Definition:Right Angle|right angles]].
+Thus, again from {{EuclidPropLink|book = XI|prop = 21|title = Solid Angle contained by Plane Angles is Less than Four Right Angles}}:
+:it is not possible to form $P$ such that its [[Definition:Vertex of Polyhedron|vertices]] are formed by $4$ [[Definition:Regular Pentagon|regular pentagons]].
+For the same reason, it is not possible to form $P$ such that its [[Definition:Vertex of Polyhedron|vertices]] are formed by more than $4$ [[Definition:Regular Pentagon|regular pentagons]].
+Hence there is only $1$ possible [[Definition:Platonic Solid|platonic solid]] whose [[Definition:Face of Polyhedron|faces]] are [[Definition:Regular Pentagon|regular pentagons]].
+$3$ [[Definition:Regular Hexagon|regular hexagons]], placed together at a [[Definition:Vertex of Polyhedron|vertex]], form $4$ [[Definition:Right Angle|right angles]].
+Thus, again from {{EuclidPropLink|book = XI|prop = 21|title = Solid Angle contained by Plane Angles is Less than Four Right Angles}}:
+:it is not possible to form $P$ such that its [[Definition:Vertex of Polyhedron|vertices]] are formed by $3$ or more [[Definition:Regular Hexagon|regular hexagons]].
+[[Definition:Regular Polygon|Regular polygons]] with more than $6$ [[Definition:Side of Polygon|sides]] have [[Definition:Vertex of Polygon|vertices]] which are greater than those of a [[Definition:Regular Hexagon|regular hexagon]].
+Therefore $3$ such [[Definition:Regular Polygon|regular polygons]], placed together at a [[Definition:Vertex of Polyhedron|vertex]], form more than $4$ [[Definition:Right Angle|right angles]].
+Thus, again from {{EuclidPropLink|book = XI|prop = 21|title = Solid Angle contained by Plane Angles is Less than Four Right Angles}}:
+:it is not possible to form $P$ such that its [[Definition:Vertex of Polyhedron|vertices]] are formed by $3$ or more [[Definition:Regular Polygon|regular polygons]] with more than $6$ [[Definition:Side of Polygon|sides]].
+Hence the $5$ possible [[Definition:Platonic Solid|platonic solids]] have been enumerated and described.
+{{qed}}
+\end{proof}
+
+\begin{proof}
+Consider a [[Definition:Convex Polyhedron|convex]] [[Definition:Regular Polyhedron|regular polyhedron]] $P$.
+Let $m$ be the number of [[Definition:Side of Polygon|sides]] of each of the [[Definition:Regular Polygon|regular polygons]] that form the [[Definition:Face of Polyhedron|faces]] of $P$.
+Let $n$ be the number of those [[Definition:Regular Polygon|polygons]] which meet at each [[Definition:Vertex of Polygon|vertex]] of $P$.
+From [[Internal Angles of Regular Polygon]], the [[Definition:Internal Angle|internal angles]] of each [[Definition:Face of Polyhedron|face]] of $P$ measure $180^\circ - \dfrac {360^\circ} m$.
+The sum of the [[Definition:Internal Angle|internal angles]] must be less than $360^\circ$.
+So:
+{{begin-eqn}}
+{{eqn | l = n \paren {180^\circ - \dfrac {360^\circ} m}
+      | o = <
+      | r = 360^\circ
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = n \paren {1 - \dfrac 2 m}
+      | o = <
+      | r = 2
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = n \paren {m - 2}
+      | o = <
+      | r = 2m
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = n \paren {m - 2} - 2 \paren {m - 2}
+      | o = <
+      | r = 2m - 2 \paren {m - 2}
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = \paren {m - 2} \paren {n - 2}
+      | o = <
+      | r = 4
+      | c = 
+}}
+{{end-eqn}}
+But $m$ and $n$ are both greater than $2$.
+So:
+:if $m = 3$, $n$ can only be $3$, $4$ or $5$
+:if $m = 4$, $n$ can only be $3$
+:if $m = 5$, $n$ can only be $3$
+and $m$ cannot be greater than $3$.
+There are $5$ possibilities in all.
+Therefore all [[Definition:Platonic Solid|platonic solids]] have been accounted for.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Space in which All Convergent Sequences have Unique Limit not necessarily Hausdorff}
+Tags: Hausdorff Spaces, Countable Complement Topology
+
+\begin{theorem}
+Let $T = \left({S, \tau}\right)$ be a [[Definition:Topological Space|topological space]].
+Let $T$ be such that all [[Definition:Convergent Sequence (Topology)|convergent sequences]] have a unique [[Definition:Limit Point of Sequence|limit point]].
+Then it is not necessarily the case that $T$ is a [[Definition:Hausdorff Space|Hausdorff space]].
+\end{theorem}
+
+\begin{proof}
+Let $T = \left({\R, \tau}\right)$ be the [[Definition:Real Number|set of real numbers]] $\R$ with the [[Definition:Countable Complement Topology|countable complement topology]].
+From [[Countable Complement Space is not T2|Countable Complement Space is not $T_2$]], $T$ is not a [[Definition:Hausdorff Space|Hausdorff space]].
+Suppose $\left\langle{x_n}\right\rangle$ is a [[Definition:Sequence|sequence in $\R$]] which converges to $x$.
+Then $C = \left\{{x_n: x_n \ne x}\right\}$ is [[Definition:Closed Set (Topology)|closed]] in $T$ because it is [[Definition:Countable Set|countable]].
+So $X \setminus C$ is a [[Definition:Neighborhood of Point|neighborhood of $x$]].
+This means there is some $N \in \N$ such that:
+:$\forall n > N: x_n \in X \setminus C$
+That is, $x_n = x$ for large $n$.
+This means that if $x_n \to y$ then $y = x$, proving [[Definition:Limit Point of Sequence|limit points]] in $T$ are unique.
+{{qed}}
+[[Category:Hausdorff Spaces]]
+[[Category:Countable Complement Topology]]
+7v3aad2g7ywuewc89zbk8mykmrsgxtu
+\end{proof}<|endoftext|>
+\section{Geometric Sequence with Coprime Extremes is in Lowest Terms}
+Tags: Geometric Sequences of Integers, Geometric Sequence with Coprime Extremes is in Lowest Terms
+
+\begin{theorem}
+Let $G_n = \sequence {a_0, a_1, \ldots, a_n}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]].
+Let:
+:$a_0 \perp a_n$
+where $\perp$ denotes [[Definition:Coprime Integers|coprimality]].
+Then $G_n$ is [[Definition:Geometric Sequence of Integers in Lowest Terms|in its lowest terms]].
+\end{theorem}
+
+\begin{proof}
+Let $G_n = \sequence {a_0, a_1, \ldots, a_n}$ be [[Definition:Natural Number|natural numbers]] in [[Definition:Geometric Sequence|geometric sequence]] such that $a_0 \perp a_n$.
+{{AimForCont}}, suppose there were to exist another set of [[Definition:Natural Number|natural numbers]] in [[Definition:Geometric Sequence|geometric sequence]]:
+:$G\,'_n = \sequence {b_0, b_1, \cdots, b_n}$
+with the same [[Definition:Common Ratio of Geometric Sequence|common ratio]] where:
+:$\forall k \in \N_{\le n}: a_k > b_k$
+From {{EuclidPropLink|book = VII|prop = 14|title = Proportion of Numbers is Transitive}}:
+:$a_0 : a_n = b_0 : b_n$
+But [[Definition:By Hypothesis|by hypothesis]]:
+:$a_0 \perp a_n$
+and so from:
+:{{EuclidPropLink|prop = 21|title = Coprime Numbers form Fraction in Lowest Terms}}
+and:
+:{{EuclidPropLink|prop = 20|title = Ratios of Fractions in Lowest Terms}}
+it follows that:
+:$a_0 \divides b_0$
+However, this contradicts the assumption that $b_0 < a_0$.
+Therefore $a_0, a_1, \cdots, a_n$ are the least of those with the same [[Definition:Common Ratio of Geometric Sequence|common ratio]].
+{{qed}}
+\end{proof}
+
+\begin{proof}
+Let $G_n = \sequence {a_0, a_1, \ldots, a_n}$ be [[Definition:Natural Number|natural numbers]] in [[Definition:Geometric Sequence|geometric sequence]] such that $a_0 \perp a_n$.
+{{AimForCont}} $G\,'_n = \sequence {b_0, b_1, \cdots, b_n}$ be another set of [[Definition:Natural Number|natural numbers]] in [[Definition:Geometric Sequence|geometric sequence]] with the same [[Definition:Common Ratio of Geometric Sequence|common ratio]] where:
+:$\forall k \in \N_{\le n}: a_k > b_k$
+By definition of [[Definition:Geometric Sequence|geometric sequence]]:
+:$a_0 = r^n a_n$
+:$b_0 = r^n b_n$
+Hence:
+:$\dfrac {a_0} {a_n} = \dfrac {b_0} {b_n}$
+[[Definition:By Hypothesis|By hypothesis]]:
+: $a_0 \perp a_n$
+Thus $\dfrac {a_0} {a_n}$ is in [[Definition:Canonical Form of Rational Number|canonical form]].
+From [[Canonical Form of Rational Number is Unique]] it follows that $a_0$ and $a_n$ are the only two [[Definition:Integer|integers]] fulfilling these conditions.
+Thus:
+:$\forall p, q: \dfrac p q = \dfrac {a_0} {a_n}: a_0 \le p, a_n \le q$
+But it was supposed that:
+:$b_0 < a_0$
+:$b_n < a_n$
+From this [[Proof by Contradiction|contradiction]] it follows that there can be no such $b_0, \ldots, b_n$.
+Hence the result.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Measurements of Common Angles/Straight Angle}
+Tags: Angles
+
+\begin{theorem}
+The measurement of a [[Definition:Straight Angle|straight angle]] is $\dfrac{360^\circ} 2 = 180^\circ$ or $\dfrac {2 \pi} 2 = \pi$.
+\end{theorem}
+
+\begin{proof}
+From [[Full Angle measures 2 Pi Radians|$2 \pi$ radians]], a full rotation is defined to be $360^\circ$ or $2 \pi$ radians.
+Since [[Definition:Straight Line|lines]] are straight, it therefore follows that from any [[Definition:Point|point]] on a line, the [[Definition:Angle|angle]] between one side of the line and the other is one half of a full rotation.
+Therefore, the measurement of a [[Definition:Straight Angle|straight angle]] is:
+:$\dfrac{360^\circ} 2 = 180^\circ$
+or:
+:$\dfrac {2 \pi} 2 = \pi$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Measurements of Common Angles/Right Angle}
+Tags: Angles
+
+\begin{theorem}
+The measurement of a [[Definition:Right Angle|right angle]] is $\dfrac {180 \degrees} 2 = 90 \degrees$ or $\dfrac \pi 2$.
+\end{theorem}
+
+\begin{proof}
+A [[Definition:Right Angle|right angle]] is equal to one half of a [[Definition:Straight Angle|straight angle]].
+From [[Measurement of Straight Angle]] it follows that the measurement of a [[Definition:Right Angle|right angle]] is $\dfrac {180 \degrees} 2 = 90 \degrees$ or $\dfrac \pi 2$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Measurements of Common Angles/Full Angle}
+Tags: Angles
+
+\begin{theorem}
+A [[Definition:Full Angle|full angle]] is equal to one full rotation.
+\end{theorem}
+
+\begin{proof}
+From [[Full Angle measures 2 Pi Radians|Full Rotation measures $2 \pi$ Radians]], a full rotation is $2 \pi$ [[Definition:Radian|radians]].
+By definition of [[Definition:Degree of Arc|degree of angle]], a full rotation is $360^\circ$.
+Therefore, a [[Definition:Full Angle|full angle]] is $360^\circ$ or $2 \pi$.
+{{qed}}
+[[Category:Angles]]
+3ohjiezeamys6ibjby32gvefryqjk7w
+\end{proof}<|endoftext|>
+\section{Measurements of Common Angles/Acute Angle}
+Tags: Angles
+
+\begin{theorem}
+An [[Definition:Acute Angle|acute angle]] measures $\theta$, where:
+:$0 \degrees < \theta < 90 \degrees$
+or:
+:$0 < \theta < \dfrac \pi 2$
+\end{theorem}
+
+\begin{proof}
+An [[Definition:Acute Angle|acute angle]] is defined to be an [[Definition:Angle|angle]] whose measure is between that of a [[Definition:Zero Angle|zero angle]] and a [[Definition:Right Angle|right angle]].
+A [[Definition:Zero Angle|zero angle]] measures $0$ by definition, and a [[Measurement of Right Angle|right angle measures $90 \degrees$ or $\dfrac \pi 2$]].
+Hence the result.
+{{qed}}
+[[Category:Angles]]
+6mq2g6a4rxr0okwgk49ktafsz00xpnl
+\end{proof}<|endoftext|>
+\section{Measurements of Common Angles/Obtuse Angle}
+Tags: Angles
+
+\begin{theorem}
+An [[Definition:Obtuse Angle|obtuse angle]] measures $\theta$, where:
+: $90^\circ < \theta < 180^\circ$
+or:
+: $\dfrac \pi 2 < \theta < \pi$
+\end{theorem}
+
+\begin{proof}
+An [[Definition:Obtuse Angle|obtuse angle]] is defined to be an [[Definition:Angle|angle]] whose measure is between that of a [[Definition:Right Angle|right angle]] and a [[Definition:Straight Angle|straight angle]].
+A [[Measurement of Right Angle|right angle measures $90^\circ$ or $\dfrac \pi 2$]] and a [[Measurement of Straight Angle|straight angle measures $180^\circ$ or $\pi$]].
+Hence the result.
+{{qed}}
+[[Category:Angles]]
+fd1dnjje8x8b0xlq1mrfiz637oxciba
+\end{proof}<|endoftext|>
+\section{Measurements of Common Angles/Reflex Angle}
+Tags: Angles
+
+\begin{theorem}
+A [[Definition:Reflex Angle|reflex angle]] measures $\theta$, where:
+: $180^\circ < \theta < 360^\circ$
+or:
+: $\pi < \theta < 2 \pi$
+\end{theorem}
+
+\begin{proof}
+A [[Definition:Reflex Angle|reflex angle]] is defined to be an [[Definition:Angle|angle]] whose measure is between that of a [[Definition:Straight Angle|straight angle]] and a [[Definition:Full Angle|full angle]].
+A [[Measurement of Straight Angle|straight angle measures $180^\circ$ or $\pi$]] and a [[Measurement of Full Angle|full angle measures $360^\circ$ or $2 \pi$]].
+Hence the result.
+{{qed}}
+[[Category:Angles]]
+6mvlsmzuva8jr6i0ng49gra5jvx9cg3
+\end{proof}<|endoftext|>
+\section{Supplementary Interior Angles implies Parallel Lines}
+Tags: Transversals (Geometry), Parallel Lines
+
+\begin{theorem}
+Given two [[Definition:Infinite Straight Line|infinite straight lines]] which are cut by a [[Definition:Transversal (Geometry)|transversal]], if the [[Definition:Interior Angle of Transversal|interior angles]] on the same side of the transversal are [[Definition:Supplementary Angles|supplementary]], then the lines are [[Definition:Parallel Lines|parallel]].
+\end{theorem}
+
+\begin{proof}
+:[[File:Parallel Cut by Transversal.png|200px]]
+Let $AB$ and $CD$ be [[Definition:Infinite Straight Line|infinite straight lines]].
+Let $EF$ be a [[Definition:Transversal (Geometry)|transversal]] that cuts them.
+Let at least one pair of [[Definition:Interior Angle of Transversal|interior angles]] on the same side of the [[Definition:Transversal (Geometry)|transversal]] be [[Definition:Supplementary Angles|supplementary]].
+{{WLOG}}, let those [[Definition:Interior Angle of Transversal|interior angles]] be $\angle BGH$ and $\angle DHG$.
+So, by definition, $\angle DHG + \angle BGH$ equals two [[Definition:Right Angle|right angles]].
+Also, from [[Two Angles on Straight Line make Two Right Angles]], $\angle AGH + \angle BGH$ equals two [[Definition:Right Angle|right angles]].
+Then from [[Axiom:Euclid's Common Notions|Euclid's first and third common notion]] and [[Axiom:Euclid's Fourth Postulate|Euclid's fourth postulate]]:
+:$\angle AGH = \angle DHG$
+Finally, by [[Equal Alternate Angles implies Parallel Lines]]:
+:$AB \parallel CD$
+{{qed}}
+{{Euclid Note|28|I|{{EuclidNoteConverse|prop = 29|title = Parallelism implies Supplementary Interior Angles|part = third}}|part = second}}
+\end{proof}<|endoftext|>
+\section{Equal Corresponding Angles implies Parallel Lines}
+Tags: Angles, Parallel Lines
+
+\begin{theorem}
+Given two [[Definition:Infinite Straight Line|infinite straight lines]] which are cut by a [[Definition:Transversal (Geometry)|transversal]], if the [[Definition:Corresponding Angles of Transversal|corresponding angles]] are equal, then the lines are [[Definition:Parallel Lines|parallel]].
+\end{theorem}
+
+\begin{proof}
+:[[File:Parallel Cut by Transversal.png|200px]]
+Let $AB$ and $CD$ be [[Definition:Infinite Straight Line|infinite straight lines]].
+Let $EF$ be a [[Definition:Transversal (Geometry)|transversal]] that cuts them.
+Let at least one pair of [[Definition:Corresponding Angles of Transversal|corresponding angles]] be equal.
+{{WLOG}}, let $\angle EGB = \angle GHD$.
+By the [[Vertical Angle Theorem]]:
+:$\angle GHD = \angle EGB = \angle AGH$
+Thus by [[Equal Alternate Angles implies Parallel Lines]]:
+:$AB \parallel CD$
+{{qed}}
+{{Euclid Note|28|I|{{EuclidNoteConverse|prop = 29|title = Parallelism implies Equal Corresponding Angles|part = second}}|part = first}}
+\end{proof}<|endoftext|>
+\section{Equivalence of Definitions of Tangent Vector}
+Tags: Differential Geometry
+
+\begin{theorem}
+Let $M$ be a [[Definition:Smooth Manifold|smooth manifold]].
+Let $m \in M$ be a [[Definition:Point|point]]. 
+Let $V$ be an [[Definition:Open Neighborhood|open neighborhood]] of $m$. 
+Let $C^\infty \left({V, \R}\right)$ be defined as the [[Definition:Set|set]] of all [[Definition:Smooth Mapping|smooth mappings]] $f: V \to \R$.
+{{TFAE|def = Tangent Vector}}
+\end{theorem}
+
+\begin{proof}
+=== Definition 2 implies Definition 1 ===
+Let $\lambda \in \R$ and $f, g \in C^\infty \left({V, \R}\right)$. 
+{{begin-eqn}}
+{{eqn | l = X_m \left({f + \lambda g} \right) 
+      | r = \frac {\mathrm d} {\mathrm d \tau} {\restriction_0} \, \left({f + \lambda g}\right) \circ \gamma \left({\tau}\right) 
+      | c = [[Definition:Tangent Vector/Definition 2|Definition 2]]
+}}
+{{eqn | r = \frac {\mathrm d} {\mathrm d \tau} {\restriction_0} \, \left({f \circ \gamma \left({\tau}\right)}\right) + \lambda \frac {\mathrm d} {\mathrm d \tau} {\restriction_0} \, \left({g \circ \gamma} \left({\tau}\right) \right) 
+}}
+{{eqn | r = X_m \left({f}\right) + \lambda \, X_m \left({g}\right) 
+}}
+{{end-eqn}}
+Thus $X_m$ is [[Definition:Linear Mapping|linear]]. 
+{{begin-eqn}}
+{{eqn | l = X_m \left({f g} \right) 
+      | r = \frac {\mathrm d} {\mathrm d \tau} {\restriction_0} \left({f g}\right) \circ \gamma \left({\tau}\right) 
+      | c = Definition 2
+}}
+{{eqn | r = \frac {\mathrm d} {\mathrm d \tau} {\restriction_0}  f \circ \gamma \left({\tau}\right) \, g \circ \gamma \left({0}\right) + f \circ \gamma \left({0}\right) \, \frac {\mathrm d} {\mathrm d \tau} {\restriction_0}  g \circ \gamma  \left({\tau}\right)
+      | c = [[Product Rule]]
+}}
+{{eqn | r = X_m \left({f}\right) \, g \left({m}\right) + f \left({m}\right) \, X_m \left({g}\right)
+      | c = Definition 2, $\gamma \left({0}\right) = m$
+}}
+{{end-eqn}}
+Hence $X_m$ satisfies the ''Leibniz law''. 
+Thus $X_m$ satisfies Definition 1.
+{{qed|lemma}}
+=== Definition 1 implies Definition 2 ===
+=== Lemma 1 === 
+Let $X_m$ be a [[Definition:Tangent Vector|tangent vector]] at $m \in M$ according to Definition 1. 
+Let $V$ be an [[Definition:Open Neighborhood|open neighborhood]] of $M$. 
+Let $f \in C^\infty \left({V, \R}\right)$ be [[Definition:Constant Mapping|constant]]. 
+Then $X_m \left({f}\right) = 0$.
+=== Proof of Lemma 1 ===
+Let $f \left({m}\right) = 0$. 
+Then, by [[Definition:Constant_Mapping|constancy]], $f = 0$ on $V$.
+Hence, by [[Definition:Linear Mapping|linearity]], $X_m \left({0}\right) = 0$. 
+Let $f \left({m}\right) \ne 0$. 
+{{begin-eqn}}
+{{eqn | l = X_m \left({f}\right)
+      | r = X_m \left({1 \, f}\right)
+}}
+{{eqn | r = X_m \left({1}\right) \, f \left({m}\right) + X_m \left({f}\right)
+      | c = Leibniz law
+}}
+{{eqn | l = \iff
+      | o =
+}}
+{{eqn | l = X_m \left({1}\right) 
+      | r = 0 
+      | c = $f \left({m}\right) \ne 0$
+}}
+{{end-eqn}}
+$f$ is [[Definition:Constant Mapping|constant]], {{iff}} $\exists \lambda \in \R : f \left({V}\right) = \left\{ {\lambda} \right\}$ {{iff}} $f = \lambda$.
+{{explain|$\exists$ in the above? Should it not be $\forall$?}}
+{{begin-eqn}}
+{{eqn | l = \lambda X_m \left({1}\right) 
+      | r = X_m  \left({\lambda}\right) 
+      | c = Definition of [[Definition:Linear Mapping|Linear Mapping]]
+}}
+{{eqn | r = X_m \left({f}\right) 
+      | c = as $f = \lambda$
+}}
+{{eqn | r = 0
+      | c = as $X_m  \left({1}\right) = 0$
+}}
+{{end-eqn}}
+{{qed|lemma}}
+Let $X_m$ be a [[Definition:Tangent Vector|tangent vector]] at $m \in M$ according to Definition 1.
+Denote $n := \dim M$. 
+Let $f \in C^\infty \left({V, \R}\right)$.
+Let $(U, \kappa)$ be a [[Definition:Chart|chart]] with $\kappa \left( {m} \right) = 0$. 
+Let $\kappa^i$ be the $i$th [[Definition:Coordinate Function|coordinate function]] of the chart $(U, \kappa)$. 
+By, [[Taylor's Theorem/n Variables]] :
+{{begin-eqn}}
+{{eqn | l = f
+      | r = f \circ \kappa^{-1} \circ \kappa
+      | c = 
+}}
+{{eqn | r = \left({f \circ \kappa^{-1} }\right) \left({\kappa}\right)
+      | c = 
+}}
+{{eqn | r = \left({f \circ \kappa^{-1} }\right) \left({0}\right) + \sum_{i \mathop = 1}^n \frac {\partial} {\left({f \circ \kappa^{-1} }\right)} {\partial{\kappa^i} } \left({0}\right) \, \kappa^i + \mathcal O_2 \left({\kappa}\right)
+      | c = 
+}}
+{{end-eqn}}
+Observe that $\displaystyle \left({f \circ \kappa^{-1} }\right) \left({0}\right) = \left({f \circ \kappa^{-1} }\right) \left( {\kappa \left({m}\right)}\right) = f \left({m}\right) $ is a [[Definition:Constant Mapping|constant mapping]] on $V$.
+{{explain|The above needs more than observation.}}
+Define $X^i := X_m \left({\kappa ^i}\right)$.
+Then by [[Definition:Linear Mapping|linearity]]:
+:$\displaystyle X_m \left({f}\right) = X_m \left({f \left({m}\right)}\right) + \sum_{i \mathop = 1}^n \frac {\partial \left({f \circ \kappa^{-1} }\right)} {\partial \kappa^i} \left({0}\right) \ X^i + X_m \left({\mathcal O_2 \left({\kappa}\right) }\right)$
+=== Lemma 2 === 
+:$X_m \left({\mathcal O_2 \left({\kappa}\right)}\right) = 0$
+=== Proof of Lemma 2 ===
+By [[Taylor's Theorem/n Variables]], for each [[Definition:Summand|summand]] of $\mathcal O _2 \left({\kappa}\right)$ there exists $i \in \left\{ {1, \ldots, n} \right\}$ and an $h \in C^\infty \left({V, \R}\right)$ with $h \left({m}\right) = 0$ such that the summand is $\kappa^i h$.
+{{begin-eqn}}
+{{eqn | l = X_m \left({\kappa^i h}\right)
+      | r = X_m \left({\kappa^i}\right) h \left({m}\right) + \kappa^i \left({m}\right) X_m \left({h}\right)
+      | c = Definition 1 (Leibniz law)
+}}
+{{eqn | r = X_m \left({\kappa^i}\right) 0 + 0 X_m \left({h}\right)
+      | c = as $h \left({m}\right) = 0$, $\kappa^i \left({m}\right) = 0$
+}}
+{{eqn | r = 0
+}}
+{{end-eqn}}
+Thus the sum $\mathcal O_2 \left({\kappa}\right)$ vanishes. 
+{{qed|lemma}}
+By Lemma 1 : 
+:$\displaystyle X_m \left({f \left({m}\right)} \right) = 0$
+By Lemma 2 : 
+:$\displaystyle X_m \left({\mathcal O_2 \left({\kappa }\right)}\right) = 0$
+Hence:
+:$\displaystyle X_m \left({f}\right) = \sum_{i \mathop = 1}^n \frac {\partial \left({f \circ \kappa^{-1}}\right)} {\partial \kappa^i} \left({0}\right) \ X^i$
+Let $\left\{{e_i}\right\}$ be a [[Definition:Basis|basis]] of $\R^n$ such that:
+:$\displaystyle \kappa = \sum_{i \mathop = 1}^n \kappa^i e_i$
+{{Disambiguate|Definition:Basis}}
+Choose a [[Definition:Smooth Curve|smooth curve]] $\gamma: I \to M$ with $0 \in I \subseteq \R$ such that $\gamma \left({0}\right) = m$ and:
+:$\dfrac {\mathrm d \kappa^i \circ \gamma} {\mathrm d \tau} \left({0}\right) := X^i$
+Then:
+{{begin-eqn}}
+{{eqn | l = X_m \left({f}\right)
+      | r = \sum_{i \mathop = 1}^n \frac{\partial \left({f \circ \kappa^{-1} }\right)} {\partial \kappa^i} \left({\kappa \left({m}\right)}\right) \frac {\mathrm d \kappa^i \circ \gamma} {\mathrm d \tau} \left({0}\right) 
+      | c = as $\kappa \left({m}\right) = 0$
+}}
+{{eqn | r = \sum_{i \mathop = 1}^n \frac {\partial \left({f \circ \kappa^{-1} }\right)} {\partial \kappa^i} \left({\kappa \circ \gamma \left({0}\right)}\right) \frac {\mathrm d \kappa^i \circ \gamma} {\mathrm d \tau} \left({0}\right) 
+      | c = as $m = \gamma \left({0}\right)$
+}}
+{{eqn | r = \sum_{i \mathop = 1}^n \left. {\frac {\partial \left({f \circ \kappa^{-1} }\right)} {\partial \kappa^i} \left({\kappa \circ \gamma \left({\tau}\right)}\right) \frac {\mathrm d \kappa^i \circ \gamma} {\mathrm d \tau} \left({\tau}\right)} \right \rvert_{\tau \mathop = 0}
+}}
+{{eqn | r = \intlimits {\frac {\mathrm d \left({f \circ \kappa^{-1} \circ \kappa \circ \gamma}\right)} {\mathrm d \tau} \left({\tau}\right)} {\tau \mathop = 0} {}
+      | c = [[Chain Rule for Derivatives]]
+}}
+{{eqn | r = \intlimits {\frac {\mathrm d \left({f \circ \gamma} \right)} {\mathrm d \tau} \left({\tau}\right)} {\tau \mathop = 0} {}
+      | c = as $f \circ \kappa^{-1} \circ \kappa = f$
+}}
+{{eqn | r = \frac {\mathrm d} {\mathrm d \tau} {\restriction_0} \, f \circ \gamma \left({\tau}\right) 
+}}
+{{end-eqn}}
+Hence $X_m$ is a [[Definition:Tangent Vector|tangent vector]] according to Definition 2. 
+This proves the assertion. 
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Parallelism implies Equal Alternate Angles}
+Tags: Transversals (Geometry), Parallel Lines
+
+\begin{theorem}
+Given two [[Definition:Infinite Straight Line|infinite straight lines]] which are cut by a [[Definition:Transversal (Geometry)|transversal]], if the lines are [[Definition:Parallel Lines|parallel]], then the [[Definition:Alternate Angles of Transversal|alternate angles]] are equal.
+\end{theorem}
+
+\begin{proof}
+:[[File:Parallel Cut by Transversal.png|200px]]
+Let $AB$ and $CD$ be [[Definition:Parallel Lines|parallel]] [[Definition:Infinite Straight Line|infinite straight lines]].
+Let $EF$ be a [[Definition:Transversal (Geometry)|transversal]] that cuts them.
+Aiming for a [[Proof by Contradiction|contradiction]], suppose the [[Definition:Alternate Angles of Transversal|alternate angles]] are not equal.
+Then one of the pair $\angle AGH$ and $\angle GHD$ must be greater.
+{{WLOG}}, let $\angle AGH$ be greater.
+From [[Two Angles on Straight Line make Two Right Angles]], $\angle AGH + \angle BGH$ equal two [[Definition:Right Angle|right angles]].
+So $\angle GHD + \angle BGH$ is less than two [[Definition:Right Angle|right angles]].
+By [[Axiom:Euclid's Fifth Postulate|Euclid's fifth postulate]], lines extended infinitely from [[Definition:Angle|angles]] less than two [[Definition:Right Angle|right angles]] must meet.
+But the lines are [[Definition:Parallel Lines|parallel]].
+So by definition the lines do not [[Definition:Intersection (Geometry)|intersect]].
+From this [[Proof by Contradiction|contradiction]] it follows that the [[Definition:Alternate Angles of Transversal|alternate angles]] are be equal.
+{{qed}}
+{{Euclid Note|29|I|{{EuclidNoteConverse|prop = 27|title = Equal Alternate Angles implies Parallel Lines}}<br><br>This is the first proposition of {{BookLink|The Elements|Euclid}} to make use of [[Axiom:Euclid's Fifth Postulate|Euclid's fifth postulate]].|part = first}}
+\end{proof}<|endoftext|>
+\section{Parallelism implies Equal Corresponding Angles}
+Tags: Transversals (Geometry), Parallel Lines
+
+\begin{theorem}
+Given two [[Definition:Infinite Straight Line|infinite straight lines]] which are cut by a [[Definition:Transversal (Geometry)|transversal]], if the lines are [[Definition:Parallel Lines|parallel]], then the [[Definition:Corresponding Angles of Transversal|corresponding angles]] are equal.
+\end{theorem}
+
+\begin{proof}
+:[[File:Parallel Cut by Transversal.png|200px]]
+Let $AB$ and $CD$ be [[Definition:Parallel Lines|parallel]] [[Definition:Infinite Straight Line|infinite straight lines]].
+Let $EF$ be a [[Definition:Transversal (Geometry)|transversal]] that cuts them.
+From [[Parallelism implies Equal Alternate Angles]]:
+:$\angle AGH = \angle DHG$
+By the [[Vertical Angle Theorem]]:
+:$\angle EGB = \angle AGH = \angle DHG$
+{{qed}}
+{{Euclid Note|29|I|{{EuclidNoteConverse|prop = 28|title = Equal Corresponding Angles implies Parallel Lines|part = first}}|part = second}}
+\end{proof}<|endoftext|>
+\section{Proportion is Reflexive}
+Tags: Proportion, Reflexive Relations
+
+\begin{theorem}
+[[Definition:Proportion|Proportion]] is a [[Definition:Reflexive Relation|reflexive relation]].
+That is, every [[Definition:Real Number|real]] [[Definition:Variable|variable]] is [[Definition:Proportional|proportional]] to itself:
+:$\forall x \in \R: x \propto x$
+\end{theorem}
+
+\begin{proof}
+Let $x$ be arbitrary.
+Then $x = 1 \times x$.
+The result follows from the definition of [[Definition:Reflexive Relation|reflexivity]] and [[Definition:Proportion|proportion]].
+{{qed}}
+[[Category:Proportion]]
+[[Category:Reflexive Relations]]
+ihv11jn67cuzil8n3v37u8hpbhzun6m
+\end{proof}<|endoftext|>
+\section{Proportion is Symmetric}
+Tags: Proportion, Symmetric Relations
+
+\begin{theorem}
+[[Definition:Proportion|Proportion]] is a [[Definition:Symmetric Relation|symmetric relation]].
+That is:
+:$\forall x, y \in \R: x \propto y \implies y \propto x$
+\end{theorem}
+
+\begin{proof}
+Let $x, y$ be arbitrary.
+Let $x$ be [[Definition:Proportional|proportional]] to $y$:
+:$x \propto y$
+Then by definition:
+{{begin-eqn}}
+{{eqn | ll= \exists k \ne 0:
+      | l = x
+      | r = k \times y
+      | c = 
+}}
+{{eqn | lll= \implies
+      | l = y
+      | r = k^{-1} \times x
+      | c = 
+}}
+{{end-eqn}}
+The result follows from the definition of [[Definition:Symmetric Relation|symmetry]] and [[Definition:Proportion|proportion]].
+{{qed}}
+[[Category:Proportion]]
+[[Category:Symmetric Relations]]
+sjurctni0ip6boyngisddlghex73gno
+\end{proof}<|endoftext|>
+\section{Proportion is Transitive}
+Tags: Proportion, Transitive Relations
+
+\begin{theorem}
+[[Definition:Proportion|Proportion]] is a [[Definition:Transitive Relation|transitive relation]].
+That is:
+:$\forall x,y,z \in \R: x \propto y \land y \propto z \implies x \propto z$
+\end{theorem}
+
+\begin{proof}
+Let $x, y, z$ be arbitrary.
+Let $x$ be [[Definition:Proportional|proportional]] to $y$ and $y$ to $z$:
+:$x \propto y \land y \propto z$
+Then by definition:
+:$\exists j, k \ne 0: x = j \times y \land y = k \times z$
+Substituting $k \times z$ for $y$:
+:$x = \left({j \times k}\right) \times z$
+so $j \times k$ is the desired [[Definition:Constant of Proportion|constant of proportion]].
+The result follows from the definition of [[Definition:Transitive Relation|transitivity]] and [[Definition:Proportion|proportion]].
+{{qed}}
+[[Category:Proportion]]
+[[Category:Transitive Relations]]
+mvflj2wl0kv6fcqafmywaghzdcg5d14
+\end{proof}<|endoftext|>
+\section{Proportion is Equivalence Relation}
+Tags: Proportion, Equivalence Relations
+
+\begin{theorem}
+[[Definition:Proportion|Proportion]] is an [[Definition:Equivalence Relation|equivalence relation]].
+\end{theorem}
+
+\begin{proof}
+* [[Proportion is Reflexive]]: $\forall x \in \R: x \propto x$
+* [[Proportion is Symmetric]]: $\forall x, y \in \R: x \propto y \implies y \propto x$
+* [[Proportion is Transitive]]: $\forall x, y, z \in \R: x \propto y \land y \propto z \implies x \propto z$
+The result follows from the definition of an [[Definition:Equivalence Relation|equivalence relation]].
+{{qed}}
+[[Category:Proportion]]
+[[Category:Equivalence Relations]]
+3vin5kgub8il4kmryq48cs3281yporg
+\end{proof}<|endoftext|>
+\section{Parallelism implies Supplementary Interior Angles}
+Tags: Transversals (Geometry), Parallel Lines
+
+\begin{theorem}
+Given two [[Definition:Infinite Straight Line|infinite straight lines]] which are cut by a [[Definition:Transversal (Geometry)|transversal]], if the lines are [[Definition:Parallel Lines|parallel]], then the [[Definition:Interior Angle of Transversal|interior angles]] on the same side of the [[Definition:Transversal (Geometry)|transversal]] are [[Definition:Supplementary Angles|supplementary]].
+\end{theorem}
+
+\begin{proof}
+:[[File:Parallel Cut by Transversal.png|200px]]
+Let $AB$ and $CD$ be [[Definition:Parallel Lines|parallel]] [[Definition:Infinite Straight Line|infinite straight lines]].
+Let $EF$ be a [[Definition:Transversal (Geometry)|transversal]] that cuts them.
+From [[Parallelism implies Equal Corresponding Angles]] and [[Axiom:Euclid's Common Notions|Euclid's second common notion]]:
+:$\angle EGB + \angle BGH = \angle DHG + \angle BGH$
+From [[Two Angles on Straight Line make Two Right Angles]], $\angle EGB + \angle BGH$ equals two right angles.
+So by definition, $\angle BGH$ and $\angle DHG$ are [[Definition:Supplementary Angles|supplementary]].
+{{qed}}
+{{Euclid Note|29|I|{{EuclidNoteConverse|prop = 28|title = Supplementary Interior Angles implies Parallel Lines|part = second}}|part = third}}
+\end{proof}<|endoftext|>
+\section{Proportion of Power}
+Tags: Proportion
+
+\begin{theorem}
+Let $x$ and $y$ be [[Definition:Proportional|proportional]].
+{{explain|Establish what types of object $x$ and $y$ are. As it stands here, they could be anything.}}
+Let $n \in \Z$. 
+Then $x^n \propto y^n$.
+\end{theorem}
+
+\begin{proof}
+Let $x \propto y$.
+Then $\exists k \ne 0: x = k \times y$ by the definition of [[Definition:Proportion|proportion]].
+Raising both sides of this equation to the [[Definition:Integer Power|$n$th power]]:
+{{begin-eqn}}
+{{eqn | l = x^n
+      | r = \left({k \times y}\right)^n
+}}
+{{eqn | r = k^n \times y^n
+}}
+{{end-eqn}}
+so $k^n$ is the desired [[Definition:Constant of Proportion|constant of proportion]].
+The result follows from the definition of [[Definition:Proportion|proportion]].
+{{qed}}
+[[Category:Proportion]]
+qpxl6b1brrbs6ujti8w4beangbxaaqz
+\end{proof}<|endoftext|>
+\section{Sine of 18 Degrees}
+Tags: Sine Function
+
+\begin{theorem}
+:$\sin 18 \degrees = \sin \dfrac \pi {10} = \dfrac {\sqrt 5 - 1} 4$
+where $\sin$ denotes the [[Definition:Sine Function|sine function]].
+\end{theorem}
+
+\begin{proof}
+From [[Sine of 90 Degrees|Sine of $90 \degrees$]]:
+:$\sin \paren {5 \times 18 \degrees} = \sin 90 \degrees = 1$.
+Consider the equation:
+: $\sin 5x = 1$
+where $x = 18 \degrees$ is one of the solutions.
+From [[Quintuple Angle Formulas/Sine|Quintuple Angle Formula of Sine]]:
+:$16 \sin^5 \theta - 20 \sin^3 \theta + 5 \sin \theta = 1$
+Let $s = \sin \theta$:
+:$16 s^5 - 20 s^3 + 5s - 1 = 0$
+That is:
+:$\paren {s - 1} \paren {4 s^2 + 2 s - 1}^2 = 0$
+Therefore, either:
+:$s = 1$
+or by the [[Quadratic Formula]]:
+:$s = \dfrac 1 4 \paren {\pm \sqrt 5 - 1}$
+Since $0 < \sin 18 \degrees < 1$:
+:$\sin 18^\circ = \dfrac {\sqrt 5 - 1} 4$
+{{qed}}
+[[Category:Sine Function]]
+q0tt2du7kh1dffh46qb53sjjb92qbxo
+\end{proof}<|endoftext|>
+\section{Cosine of 72 Degrees}
+Tags: Cosine Function, Cosine of 72 Degrees
+
+\begin{theorem}
+:$\cos 72 \degrees = \cos \dfrac {2 \pi} 5 = \dfrac {\sqrt 5 - 1} 4$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \cos 72 \degrees
+      | r = \map \cos {90 \degrees - 18 \degrees}
+}}
+{{eqn | r = \sin 18 \degrees
+      | c = [[Cosine of Complement equals Sine]]
+}}
+{{eqn | r = \dfrac {\sqrt 5 - 1} 4
+      | c = [[Sine of 18 Degrees|Sine of $18 \degrees$]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \cos 72 \degrees
+      | r = 2 \cos 36 \degrees - 1
+}}
+{{eqn | r = 2 \paren {\dfrac \phi 2}^2 - 1
+      | c = [[Cosine of 36 Degrees|Cosine of $36 \degrees$]]
+}}
+{{eqn | r = \dfrac {\phi^2} 2 - 1
+      | c = 
+}}
+{{eqn | r = \dfrac {\phi + 1} 2 - 1
+      | c = [[Square of Golden Mean equals One plus Golden Mean]]
+}}
+{{eqn | r = \dfrac {\phi - 1} 2
+      | c = 
+}}
+{{eqn | r = -\dfrac {1 - \phi} 2
+      | c = 
+}}
+{{eqn | r = \dfrac {\phi^{-1} } 2
+      | c = [[Reciprocal Form of One Minus Golden Mean]]
+}}
+{{eqn | r = \dfrac {\sqrt 5 - 1} 4
+      | c = {{Defof|Golden Mean|index = 2}}, and algebra
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Sine of 72 Degrees}
+Tags: Sine Function
+
+\begin{theorem}
+:$\sin 72 \degrees = \sin \dfrac {2 \pi} 5 = \dfrac {\sqrt{10 + 2 \sqrt 5} } 4$
+where $\sin$ denotes the [[Definition:Sine Function|sine function]].
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \sin 72 \degrees
+      | r = \sqrt {1 - \cos^2 72 \degrees}
+      | c = [[Sum of Squares of Sine and Cosine]]
+}}
+{{eqn | r = \sqrt {1 - \paren {\dfrac{\sqrt 5 - 1} 4}^2}
+      | c = [[Cosine of 72 Degrees|Cosine of $72 \degrees$]]
+}}
+{{eqn | r = \sqrt {1 - \dfrac {6 - 2 \sqrt 5} {16} }
+}}
+{{eqn | r = \dfrac {\sqrt {10 + 2 \sqrt 5} } 4
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Cosine of 18 Degrees}
+Tags: Cosine Function
+
+\begin{theorem}
+:$\cos 18^\circ = \cos \dfrac \pi {10} = \dfrac {\sqrt{10 + 2 \sqrt 5}} 4$
+where $\cos$ denotes the [[Definition:Cosine Function|cosine function]].
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \cos 18^\circ
+      | r = \cos \left({90^\circ - 72^\circ}\right)
+}}
+{{eqn | r = \sin 72^\circ
+      | c = [[Cosine of Complement equals Sine]]
+}}
+{{eqn | r = \frac {\sqrt {10 + 2 \sqrt 5} } 4
+      | c = [[Sine of 72 Degrees]]
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Cosine Function]]
+qfdwbrom2u1wtnh06j1q6ipf4ebudx0
+\end{proof}<|endoftext|>
+\section{Sine of 3 Degrees}
+Tags: Sine Function
+
+\begin{theorem}
+:$\sin 3^\circ = \sin \dfrac \pi {60} = \dfrac {\sqrt{30} + \sqrt{10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } {16}$
+where $\sin$ denotes the [[Definition:Sine Function|sine function]].
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \sin 3^\circ
+      | r = \sin \left({75^\circ - 72^\circ}\right)
+}}
+{{eqn | r = \sin 75^\circ \cos 72^\circ - \cos 75^\circ \sin 72^\circ
+      | c = [[Sine of Difference]]
+}}
+{{eqn | r = \dfrac {\sqrt 6 + \sqrt 2} 4 \times \dfrac {\sqrt 5 - 1} 4 - \cos 75^\circ \sin 72^\circ
+      | c = [[Sine of 75 Degrees|Sine of $75^\circ$]], [[Cosine of 72 Degrees|Cosine of $72^\circ$]] ...
+}}
+{{eqn | r = \dfrac {\sqrt 6 + \sqrt 2} 4 \times \dfrac {\sqrt 5 - 1} 4 - \dfrac {\sqrt 6 - \sqrt 2} 4 \times \dfrac {\sqrt {10 + 2 \sqrt 5} } 4
+      | c = ... [[Cosine of 75 Degrees|Cosine of $75^\circ$]], [[Sine of 72 Degrees|Sine of $72^\circ$]]
+}}
+{{eqn | r = \dfrac 1 {16} \times \left({\left({\sqrt 6 + \sqrt 2}\right) \times \left({\sqrt 5 - 1}\right) - \left({\sqrt 6 - \sqrt 2}\right) \times \left({\sqrt {10 + 2 \sqrt 5} }\right) }\right)
+}}
+{{eqn | r = \dfrac 1 {16} \times \left({\left({\sqrt{30} + \sqrt{10} - \sqrt 6 - \sqrt 2}\right) - \left({\sqrt {60 + 12 \sqrt 5} - \sqrt {20 + 4 \sqrt 5} }\right) }\right)
+}}
+{{eqn | r = \dfrac 1 {16} \times \left({\sqrt{30} + \sqrt{10} - \sqrt 6 - \sqrt 2 - \sqrt {60 + 12 \sqrt5 } + \sqrt {20 + 4 \sqrt 5} }\right)
+}}
+{{eqn | r = \dfrac 1 {16} \times \left({\sqrt{30} + \sqrt{10} - \sqrt 6 - \sqrt 2 - 2\sqrt {15 + 3 \sqrt5 } + 2 \sqrt {5 + \sqrt 5} }\right)
+}}
+{{eqn | r = \dfrac {\sqrt{30} + \sqrt{10} - \sqrt 6 - \sqrt 2 - 2 \sqrt {15 + 3 \sqrt 5} + 2\sqrt {5 + \sqrt 5} } {16}
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Sine Function]]
+kjgsyns60rpt2i9bn4jxxxy59rzkp3s
+\end{proof}<|endoftext|>
+\section{Cosine of 3 Degrees}
+Tags: Cosine Function
+
+\begin{theorem}
+:$\cos 3^\circ = \cos \dfrac {\pi} {60} = \dfrac{ \sqrt{30} - \sqrt{10} - \sqrt 6 + \sqrt 2 + 2 \sqrt{15 + 3 \sqrt 5} + 2 \sqrt{5 + \sqrt 5} } {16}$
+where $\cos$ denotes the [[Definition:Cosine Function|cosine function]].
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \cos 3^\circ
+      | r = \cos \left({75^\circ - 72^\circ}\right)
+}}
+{{eqn | r = \cos 75^\circ \cos 72^\circ + \sin 75^\circ \sin 72^\circ
+      | c = [[Cosine of Difference]]
+}}
+{{eqn | r = \dfrac {\sqrt 6 - \sqrt 2} 4 \times \dfrac {\sqrt 5 - 1} 4 + \dfrac {\sqrt 6 + \sqrt 2} 4 \times \dfrac {\sqrt {10 + 2 \sqrt5 } } 4
+      | c = [[Cosine of 75 Degrees]], [[Cosine of 72 Degrees]], [[Sine of 75 Degrees]], [[Sine of 72 Degrees]]
+}}
+{{eqn | r = \dfrac 1 {16} \times \left[{\left({\sqrt 6 - \sqrt 2}\right) \times \left({\sqrt 5 - 1}\right) + \left({\sqrt 6 + \sqrt 2}\right) \times \left({\sqrt {10 + 2 \sqrt 5} }\right) }\right]
+}}
+{{eqn | r = \dfrac 1 {16} \times \left[{\left({\sqrt{30} - \sqrt{10} - \sqrt 6 + \sqrt 2}\right) + \left({\sqrt{60 + 12 \sqrt 5} + \sqrt{20 + 4 \sqrt 5} }\right) }\right]
+}}
+{{eqn | r = \dfrac 1 {16} \times \left({\sqrt{30} - \sqrt{10} - \sqrt 6 + \sqrt 2 + \sqrt{60 + 12 \sqrt 5} + \sqrt{20 + 4 \sqrt 5} }\right)
+}}
+{{eqn | r = \dfrac 1 {16} \times \left({\sqrt{30} - \sqrt{10} - \sqrt 6 + \sqrt 2 + 2 \sqrt{15 + 3 \sqrt 5} + 2 \sqrt{5 + \sqrt 5} }\right)
+}}
+{{eqn | r = \dfrac{\sqrt{30} - \sqrt{10} - \sqrt 6 + \sqrt 2 + 2 \sqrt{15 + 3 \sqrt 5} + 2 \sqrt{5 + \sqrt 5} }{16}
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Cosine Function]]
+fl2cb66is22nvq38tpw30zkxhq1vq57
+\end{proof}<|endoftext|>
+\section{Square Root of Sum as Sum of Square Roots}
+Tags: Algebra, Square Root of Sum as Sum of Square Roots
+
+\begin{theorem}
+Let $a, b \in \R, a \ge b$.
+Then:
+:$\sqrt {a + b} = \sqrt {\dfrac a 2 + \dfrac {\sqrt {a^2 - b^2}} 2} + \sqrt {\dfrac a 2 - \dfrac {\sqrt {a^2 - b^2}} 2}$
+\end{theorem}
+
+\begin{proof}
+Let $\sqrt {a + b}$ be expressed in the form $\sqrt c + \sqrt d$.
+From [[Square of Sum]]:
+: $a + b = c + d + 2 \sqrt {c d}$
+We now need to solve the [[Definition:Simultaneous Equations|simultaneous equations]]:
+: $a = c + d$
+: $b = 2 \sqrt {c d}$
+First:
+{{begin-eqn}}
+{{eqn | l = a
+      | r = c + d
+}}
+{{eqn | n = 1
+      | ll= \leadsto
+      | l = d
+      | r = a - c
+      | c = subtracting $c$ from both sides
+}}
+{{end-eqn}}
+Solving for $c$:
+{{begin-eqn}}
+{{eqn | l = b
+      | r = 2 \sqrt {c d}
+}}
+{{eqn | ll= \leadsto
+      | l = b^2
+      | r = 4 c d
+      | c = [[Definition:Squaring|squaring]] both sides
+}}
+{{eqn | r = 4 c \paren {a - c}
+      | c = substituting $d = a - c$ from $(1)$
+}}
+{{eqn | r = 4 a c - 4 c^2
+      | c = [[Real Multiplication Distributes over Addition]]
+}}
+{{eqn | ll= \leadsto
+      | l = 4 c^2 - 4 a c + b^2
+      | r = 0
+      | c = adding $4 c^2 - 4 a c$ to both sides
+}}
+{{eqn | n = 2
+      | ll= \leadsto
+      | l = c
+      | r = \frac a 2 \pm \frac {\sqrt {a^2 - b^2} } 2
+      | c = [[Quadratic Formula]]
+}}
+{{end-eqn}}
+Solving for $d$:
+{{begin-eqn}}
+{{eqn | l = d
+      | r = a - c
+}}
+{{eqn | r = a - \frac a 2 \mp \frac {\sqrt {a^2 - b^2} } 2
+      | c = substituting $c = \dfrac a 2 \pm \dfrac {\sqrt {a^2 - b^2} } 2$ from $(2)$
+}}
+{{eqn | r = \frac a 2 \mp \frac {\sqrt {a^2 - b^2} } 2
+}}
+{{end-eqn}}
+From [[Real Addition is Commutative]], the sign of the [[Definition:Square Root|square root]] may be chosen arbitrarily, provided opposite signs are chosen for $c$ and $d$.
+{{qed}}
+\end{proof}
+
+\begin{proof}
+From [[Sum of Square Roots as Square Root of Sum]]:
+:$\sqrt p + \sqrt q = \sqrt {p + q + \sqrt {4pq}}$
+Let
+:$p = \dfrac a 2 + \dfrac {\sqrt {a^2 - b^2}} 2$,
+:$q = \dfrac a 2 - \dfrac {\sqrt {a^2 - b^2}} 2$.
+Then
+{{begin-eqn}}
+{{eqn | l = p + q
+      | r = \frac a 2 + \frac {\sqrt {a^2 - b^2} } 2 + \frac a 2 - \frac {\sqrt {a^2 - b^2} } 2
+}}
+{{eqn | r = a
+}}
+{{end-eqn}}
+{{begin-eqn}}
+{{eqn | l = \sqrt {4pq}
+      | r = \sqrt {4 \paren {\frac a 2 + \frac {\sqrt {a^2 - b^2} } 2} \paren {\frac a 2 - \frac {\sqrt {a^2 - b^2} } 2} }
+}}
+{{eqn | r = \sqrt {\paren {a + \sqrt {a^2 - b^2} } \paren {a - \sqrt {a^2 - b^2} } }
+      | c = [[Real Multiplication Distributes over Addition]]
+}}
+{{eqn | r = \sqrt {a^2 - \paren {a^2 - b^2} }
+      | c = [[Difference of Two Squares]]
+}}
+{{eqn | r = \sqrt {b^2}
+}}
+{{eqn | r = b
+}}
+{{end-eqn}}
+Therefore:
+:$\sqrt {\dfrac a 2 + \dfrac {\sqrt {a^2 - b^2}} 2} + \sqrt {\dfrac a 2 - \dfrac {\sqrt {a^2 - b^2}} 2} = \sqrt {a + b}$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Sine of 1 Degree}
+Tags: Sine Function
+
+\begin{theorem}
+:$\sin 1^\circ = \sin \dfrac {\pi} {180} = $
+where $\sin$ denotes the [[Definition:Sine Function|sine function]].
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \sin \left({3 \times 1^\circ}\right)
+      | r = 3 \sin 1^\circ - 4 \sin^3 1^\circ
+      | c = [[Triple Angle Formula for Sine]]
+}}
+{{eqn | l = \sin 3^\circ
+      | r = 3 \sin 1^\circ - 4 \sin^3 1^\circ
+}}
+{{eqn | l = 4 \sin^3 1^\circ - 3 \sin 1^\circ + \sin 3^\circ
+      | r = 0
+}}
+{{end-eqn}}
+This is in the form:
+:$a x^3 + b x^2 + c x + d = 0$
+where:
+: $x = \sin 1^\circ$
+: $a = 4$
+: $b = 0$
+: $c = -3$
+: $d = \sin 3^\circ$
+From [[Cardano's Formula]]:
+:$x = S + T$
+where:
+:$S = \sqrt [3] {R + \sqrt{Q^3 + R^2}}$
+:$T = \sqrt [3] {R - \sqrt{Q^3 + R^2}}$
+where:
+{{begin-eqn}}
+{{eqn | l = Q
+      | r = \dfrac {3 a c - b^2} {9 a^2}
+      | c = 
+}}
+{{eqn | r = \dfrac {3 \times 4 \times \left({-3}\right) - 0^2} {9 \times 4^2}
+      | c = 
+}}
+{{eqn | r = -\dfrac 1 4
+      | c = 
+}}
+{{end-eqn}}
+and:
+{{begin-eqn}}
+{{eqn | l = R
+      | r = \dfrac {9 a b c - 27 a^2 d - 2 b^3} {54 a^3}
+      | c = 
+}}
+{{eqn | r = \dfrac {9 \times 4 \times 0 \times \left({-3}\right) - 27 \times 4^2 \times \sin 3^\circ - 2 \times 0^3} {54 \times 4^3}
+      | c = 
+}}
+{{eqn | r = \dfrac {\sin 3^\circ} 8
+      | c = 
+}}
+{{end-eqn}}
+Thus:
+{{begin-eqn}}
+{{eqn | l = \sin 1^\circ
+      | r = S + T
+      | c = putting $x = S + T$
+}}
+{{eqn | r = \sqrt[3] {R + \sqrt{Q^3 + R^2} } + \sqrt[3] {R - \sqrt{Q^3 + R^2} }
+      | c = substituting for $S$ and $T$
+}}
+{{eqn | r = \sqrt[3] {R + \sqrt{-\left({\dfrac 4 9}\right)^3 + R^2} } + \sqrt[3] {R - \sqrt {-\left({\dfrac 4 9}\right)^3 + R^2} }
+      | c = substituting for $Q$
+}}
+{{eqn | r = \sqrt[3] {-\dfrac{\sin 3^\circ} 8 + \sqrt{-\left({ \dfrac 4 9}\right)^3 + \dfrac {\sin^2 3^\circ} {64} } }
+      | c = substituting for $R$
+}}
+{{eqn | o = 
+      | ro= +
+      | r = \sqrt[3] {-\dfrac {\sin 3^\circ} 8 - \sqrt{-\left({ \dfrac 4 9}\right)^3 + \dfrac {\sin^2 3^\circ} {64} } }
+      | c = 
+}}
+{{end-eqn}}
+{{mistake|I believe the substituted value for $Q$ above may be wrong. It appears that $Q {{=}} -\dfrac 4 9$ has been used, but in the calculation above it may be the case that it should be $Q {{=}} -\dfrac 1 4$. The algebra has gone wrong somewhere and needs to be reviewed.}}
+By [[Sine of 3 Degrees]]:
+{{begin-eqn}}
+{{eqn | l = \sin 1^\circ
+      | r = \sqrt[3] {-\dfrac{ \left({ \dfrac{ \sqrt{30} + \sqrt{10} - \sqrt 6 - \sqrt 2 - 2\sqrt {15 + 3 \sqrt5 } + 2\sqrt {5 + \sqrt5 } }{16} }\right) }{8} + \sqrt{-\left({ \dfrac{4}{9} }\right)^3 + \dfrac{ \left({ \dfrac{ \sqrt{30} + \sqrt{10} - \sqrt 6 - \sqrt 2 - 2\sqrt {15 + 3 \sqrt5 } + 2\sqrt {5 + \sqrt5 } }{16} }\right)^2 }{64} } }
+}}
+{{eqn | o = 
+      | ro= +
+      | r = \sqrt[3] {-\dfrac{ \left({ \dfrac{ \sqrt{30} + \sqrt{10} - \sqrt 6 - \sqrt 2 - 2\sqrt {15 + 3 \sqrt5 } + 2\sqrt {5 + \sqrt5 } }{16} }\right) }{8} - \sqrt{-\left({ \dfrac{4}{9} }\right)^3 + \dfrac{\left({ \dfrac{ \sqrt{30} + \sqrt{10} - \sqrt 6 - \sqrt 2 - 2\sqrt {15 + 3 \sqrt5 } + 2\sqrt {5 + \sqrt5 } }{16} }\right)^2}{64} } }
+}}
+{{eqn | r = \sqrt[3] {-\dfrac{ \left({ \dfrac{ \sqrt{30} + \sqrt{10} - \sqrt 6 - \sqrt 2 - 2\sqrt {15 + 3 \sqrt5 } + 2\sqrt {5 + \sqrt5 } }{16} }\right) }{8} + \sqrt{-\left({ \dfrac{4}{9} }\right)^3 + \dfrac{ \left({ \dfrac{-8 + 24\sqrt3 - 16\sqrt5 - 8\sqrt{15} - 8\sqrt {50 + 10\sqrt5} + 8\sqrt {10 + 2 \sqrt5} - 8\sqrt{90 + 30\sqrt5} }{16^2} }\right) }{64} } }
+}}
+{{eqn | o = 
+      | ro= +
+      | r = \sqrt[3] {-\dfrac{ \left({ \dfrac{ \sqrt{30} + \sqrt{10} - \sqrt 6 - \sqrt 2 - 2\sqrt {15 + 3 \sqrt5 } + 2\sqrt {5 + \sqrt5 } }{16} }\right) }{8} - \sqrt{-\left({ \dfrac{4}{9} }\right)^3 + \dfrac{\left({ \dfrac{-8 + 24\sqrt3 - 16\sqrt5 - 8\sqrt{15} - 8\sqrt {50 + 10\sqrt5} + 8\sqrt {10 + 2 \sqrt5} - 8\sqrt{90 + 30\sqrt5} }{16^2} }\right)}{64} } }
+}}
+{{eqn | r = \sqrt[3] {-\dfrac{ \sqrt{30} + \sqrt{10} - \sqrt 6 - \sqrt 2 - 2\sqrt {15 + 3 \sqrt5 } + 2\sqrt {5 + \sqrt5 } } {128} + \sqrt{-\dfrac{64}{729} + \dfrac{ -8 + 24\sqrt3 - 16\sqrt5 - 8\sqrt{15} - 8\sqrt {50 + 10\sqrt5} + 8\sqrt {10 + 2 \sqrt5} - 8\sqrt{90 + 30\sqrt5} }{16384} } }
+}}
+{{eqn | o = 
+      | ro= +
+      | r = \sqrt[3] {-\dfrac{ \sqrt{30} + \sqrt{10} - \sqrt 6 - \sqrt 2 - 2\sqrt {15 + 3 \sqrt5 } + 2\sqrt {5 + \sqrt5 } } {128} - \sqrt{-\dfrac{64}{729} + \dfrac{ -8 + 24\sqrt3 - 16\sqrt5 - 8\sqrt{15} - 8\sqrt {50 + 10\sqrt5} + 8\sqrt {10 + 2 \sqrt5} - 8\sqrt{90 + 30\sqrt5} }{16384} } }
+}}
+{{eqn | r = \sqrt[3] {-\dfrac{ \sqrt{30} + \sqrt{10} - \sqrt 6 - \sqrt 2 - 2\sqrt {15 + 3 \sqrt5 } + 2\sqrt {5 + \sqrt5 } } {128} + \sqrt{-\dfrac{64}{729} + \dfrac{ -1 + 3\sqrt3 - 2\sqrt5 - \sqrt{15} - \sqrt {50 + 10\sqrt5} + \sqrt {10 + 2 \sqrt5} - \sqrt{90 + 30\sqrt5} }{2048} } }
+}}
+{{eqn | o = 
+      | ro= +
+      | r = \sqrt[3] {-\dfrac{ \sqrt{30} + \sqrt{10} - \sqrt 6 - \sqrt 2 - 2\sqrt {15 + 3 \sqrt5 } + 2\sqrt {5 + \sqrt5 } } {128} - \sqrt{-\dfrac{64}{729} + \dfrac{ -1 + 3\sqrt3 - 2\sqrt5 - \sqrt{15} - \sqrt {50 + 10\sqrt5} + \sqrt {10 + 2 \sqrt5} - \sqrt{90 + 30\sqrt5} }{2048} } }
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Sine Function]]
+hhuqg80tw9ly32lbnlu5xcnr2qlgnct
+\end{proof}<|endoftext|>
+\section{Rectangle is Parallelogram}
+Tags: Rectangles, Parallelograms
+
+\begin{theorem}
+Let $ABCD$ be a [[Definition:Rectangle|rectangle]].
+Then $ABCD$ is a [[Definition:Parallelogram|parallelogram]].
+\end{theorem}
+
+\begin{proof}
+Let $ABCD$ be a [[Definition:Rectangle|rectangle]].
+[[Proof by Contradiction|Seeking a contradiction]], assume $ABCD$ is ''not'' a [[Definition:Parallelogram|parallelogram]].
+{{image wanted|PW gives an error message when uploading images. An email has been sent to Joe.}}
+{{WLOG}} let [[Definition:Line Segment|line segments]] $AD$ and $BC$ ''not'' be parallel.
+Then [[Axiom:Euclid's Second Postulate|extend $AD$ and $BC$]] into two [[Definition:Line/Infinite Straight Line|infinite straight lines]].
+By the [[Axiom:Parallel Postulate|Parallel Postulate]], the lines will eventually meet at one side or the other.
+Let their point of [[Definition:Intersection (Geometry)|intersection]] be $E$.
+There are two possibilities:
+:$ABE$ is a [[Definition:Triangle (Geometry)|triangle]]
+:$CDE$ is a [[Definition:Triangle (Geometry)|triangle]].
+{{WLOG}}, let $ABE$ be a triangle.
+If $ABE$ is a triangle, then by [[Sum of Angles of Triangle equals Two Right Angles]], $\angle ABE + \angle BEA + \angle EAB$ add to two [[Definition:Right Angle|right angles]].
+But by the [[Definition:Rectangle|definition of a rectangle]], $\angle ABE + \angle EAB$ add to two right angles.
+As $\angle BEA$ is not a [[Definition:Zero Angle|zero angle]] (else $ABD$ would not be a triangle), this cannot be so.
+We are forced to conclude that that $ABCD$ is not a rectangle, a [[Definition:Contradiction|contradiction]].
+{{qed}}
+[[Category:Rectangles]]
+[[Category:Parallelograms]]
+qiwnxy6fxk3n89o4f39a7c37oltaget
+\end{proof}<|endoftext|>
+\section{Condition for Denesting of Square Root}
+Tags: Square Roots, Condition for Denesting of Square Root
+
+\begin{theorem}
+Let $a, b \in \Q_{\ge 0}$.
+Suppose $\sqrt b \notin \Q$.
+Then:
+:$\exists p, q \in \Q: \sqrt {a + \sqrt b} = \sqrt p + \sqrt q$
+{{iff}}:
+:$\exists n \in \Q: a^2 - b = n^2$.
+\end{theorem}
+
+\begin{proof}
+=== [[Condition for Denesting of Square Root/Lemma|Lemma]] ===
+{{:Condition for Denesting of Square Root/Lemma}}{{qed|lemma}}
+=== Necessary condition ===
+{{begin-eqn}}
+{{eqn | l = \sqrt {a + \sqrt b}
+      | r = \sqrt p + \sqrt q
+}}
+{{eqn | r = \sqrt {p + q + \sqrt {4 p q} }
+      | c = [[Sum of Square Roots as Square Root of Sum]]
+}}
+{{eqn | ll= \leadsto
+      | l = a
+      | r = p + q
+      | c = by [[Condition for Denesting of Square Root/Lemma|Lemma]]
+}}
+{{eqn | l = b
+      | r = 4 p q
+      | c = by [[Condition for Denesting of Square Root/Lemma|Lemma]]
+}}
+{{eqn | ll= \leadsto
+      | l = 0
+      | r = b - 4 p q
+}}
+{{eqn | r = b - 4 p \paren {a - p}
+}}
+{{eqn | r = b - 4 a p + 4 p^2
+}}
+{{eqn | ll= \leadsto
+      | l = p
+      | r = \frac {4 a \pm \sqrt {16 a^2 - 16 b} } 8
+      | c = [[Quadratic Formula]]
+}}
+{{eqn | r = \frac a 2 \pm \frac {\sqrt {a^2 - b} } 2
+}}
+{{end-eqn}}
+This shows that, in order for $p \in \Q$, we must have $\sqrt {a^2 - b} \in \Q$ as well.
+The result follows from $\paren {\sqrt {a^2 - b} }^2 = a^2 - b$.
+{{qed|lemma}}
+=== Sufficient condition ===
+Let $n^2 = a^2 - b$.
+As $a^2 = b + n^2$ it follows that:
+:$a \ge \sqrt b$
+Then:
+{{begin-eqn}}
+{{eqn | l = \sqrt {a + \sqrt b}
+      | r = \sqrt {\dfrac a 2 + \dfrac {\sqrt {a^2 - b} } 2} + \sqrt {\dfrac a 2 - \dfrac {\sqrt {a^2 - b} } 2}
+      | c = [[Square Root of Sum as Sum of Square Roots]]
+}}
+{{eqn | r = \sqrt {\dfrac a 2 + \dfrac {\sqrt {n^2} } 2} + \sqrt {\dfrac a 2 - \dfrac {\sqrt {n^2} } 2}
+      | c = substituting $n^2 = a^2 - b$
+}}
+{{eqn | r = \sqrt {\dfrac a 2 + \dfrac n 2} + \sqrt {\dfrac a 2 - \dfrac n 2}
+}}
+{{end-eqn}}
+By [[Rational Addition is Closed]] and [[Rational Subtraction is Closed]]:
+:$\dfrac a 2 + \dfrac n 2$ and $\dfrac a 2 - \dfrac n 2$
+are [[Definition:Rational Number|rational]].
+{{qed}}
+[[Category:Square Roots]]
+[[Category:Condition for Denesting of Square Root]]
+3sarxzk03ddhfc97y129a34h15q2tui
+\end{proof}<|endoftext|>
+\section{Line at Right Angles to Diameter of Circle/Porism}
+Tags: Circles
+
+\begin{theorem}
+{{:Euclid:Proposition/III/16/Porism}}
+\end{theorem}
+
+\begin{proof}
+Follows directly from [[Line at Right Angles to Diameter of Circle]].
+\end{proof}<|endoftext|>
+\section{Characteristic Function of Union/Variant 1}
+Tags: Characteristic Function of Union
+
+\begin{theorem}
+:$\chi_{A \mathop \cup B} = \min \set {\chi_A + \chi_B, 1}$
+\end{theorem}
+
+\begin{proof}
+By [[Characteristic Function Determined by 1-Fiber]], it suffices to show:
+:$\min \set {\map {\chi_A} s + \map {\chi_B} s, 1} = 1 \iff s \in A \cup B$
+By the nature of the [[Definition:Min Operation|min operation]], this amounts to showing that:
+:$\map {\chi_A} s + \map {\chi_B} s \ge 1 \iff s \in A \cup B$
+As $\chi_A, \chi_B$ are [[Definition:Characteristic Function of Set|characteristic functions]], the left-hand side amounts to:
+:$s \in A \lor s \in B$
+which is precisely the [[Definition:Set Union|definition of $s \in A \cup B$]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Characteristic Function of Union/Variant 2}
+Tags: Characteristic Function of Union
+
+\begin{theorem}
+:$\chi_{A \mathop \cup B} = \chi_A + \chi_B - \chi_{A \mathop \cap B}$
+\end{theorem}
+
+\begin{proof}
+From [[Subset of Union]]:
+:$A, B \subseteq A \cup B$
+From [[Intersection is Subset of Union]]:
+:$A \cap B \subseteq A \cup B$
+Thus from [[Characteristic Function of Subset]]:
+:$\map {\chi_{A \mathop \cup B} } s = 0 \implies \map {\chi_A} s = \map {\chi_B} s = \map {\chi_{A \mathop \cap B} } s = 0$
+Now suppose that $\map {\chi_A} s + \map {\chi_B} s - \map {\chi_{A \mathop \cap B} } s = 0$.
+That is:
+:$\map {\chi_A} s + \map {\chi_B} s = \map {\chi_{A \mathop \cap B} } s$
+Suppose the latter equals $1$, that is $s \in A \cap B$.
+Then both $s \in A$ and $s \in B$, so by definition of [[Definition:Characteristic Function of Set|characteristic function]]:
+:$\map {\chi_A} s + \map {\chi_B} s = 1 + 1 = 2$
+Since $2 \ne 1$, it follows that $\map {\chi_{A \mathop \cap B} } s \ne 1$, that is it equals $0$.
+Thence, it follows that:
+:$\map {\chi_A} s + \map {\chi_B} s = 0$.
+This only happens when $\map {\chi_A} s = \map {\chi_B} s = 0$.
+Thus, $s \notin A$ and $s \notin B$, so $s \notin A \cup B$.
+It finally follows that $\map {\chi_{A \mathop \cup B} } s = 0$.
+It is now established that:
+:$\map {\chi_{A \mathop \cup B} } s = 0 \iff \map {\chi_A} s + \map {\chi_B} s - \map {\chi_{A \mathop \cap B} } s = 0$
+and from [[Characteristic Function Determined by 0-Fiber]], the result follows.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Characteristic Function of Union/Variant 3}
+Tags: Characteristic Function of Union
+
+\begin{theorem}
+:$\chi_{A \mathop \cup B} = \max \set {\chi_A, \chi_B}$
+\end{theorem}
+
+\begin{proof}
+Suppose $\map {\chi_{A \mathop \cup B} } s = 0$.
+Then $s \notin A \cup B$, so $s \notin A$ and $s \notin B$.
+Hence:
+:$\map {\chi_A} s = \map {\chi_B} s = 0$
+and by definition of [[Definition:Max Operation|max operation]]:
+:$\max \set {\map {\chi_A} s, \map {\chi_B} s} = 0$
+Conversely, suppose:
+:$\max \set {\map {\chi_A} s, \map {\chi_B} s} = 0$
+Then it follows that:
+:$\map {\chi_A} s = \map {\chi_B} s = 0$
+because [[Definition:Characteristic Function of Set|characteristic functions]] are $0$ or $1$.
+Hence $s \notin A$ and $s \notin B$, so $s \notin A \cup B$.
+That is:
+:$\map {\chi_{A \mathop \cup B} } s = 0$
+Above considerations give:
+:$\map {\chi_{A \mathop \cup B} } s = 0 \iff \max \set {\map {\chi_A} s, \map {\chi_B} s} = 0$
+and applying [[Characteristic Function Determined by 0-Fiber]], the result follows.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Characteristic Function of Intersection/Variant 1}
+Tags: Characteristic Functions, Set Intersection
+
+\begin{theorem}
+:$\chi_{A \cap B} = \chi_A \chi_B$
+\end{theorem}
+
+\begin{proof}
+By [[Characteristic Function Determined by 1-Fiber]], it suffices to show that:
+:$\map {\chi_A} s \, \map {\chi_B} s = 1 \iff s \in A \cap B$
+Now, both $\chi_A$ and $\chi_B$ are [[Definition:Characteristic Function of Set|characteristic functions]].
+It follows that, for any $s \in S$:
+:$\map {\chi_A} s \, \map {\chi_B} s = 1 \iff \map {\chi_A} s = \map {\chi_B} s = 1$
+By definition of $\chi_A$ and $\chi_B$, this is equivalent to the statement that both $s \in A$ and $s \in B$.
+That is, $s \in A \cap B$, by definition of [[Definition:Set Intersection|set intersection]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Characteristic Function of Intersection/Variant 2}
+Tags: Characteristic Functions, Set Intersection
+
+\begin{theorem}
+:$\chi_{A \cap B} = \min \left\{{\chi_A, \chi_B}\right\}$
+\end{theorem}
+
+\begin{proof}
+By [[Characteristic Function Determined by 1-Fiber]], it suffices to show that:
+:$\min \left\{{\chi_A \left({s}\right), \chi_B \left({s}\right)}\right\} = 1 \iff s \in A \cap B$
+By definition of [[Definition:Characteristic Function of Set|characteristic function]], we have:
+:$\min \left\{{\chi_A \left({s}\right), \chi_B \left({s}\right)}\right\} = 1$
+{{iff}} $\chi_A \left({s}\right) = \chi_B \left({s}\right) = 1$ because $\chi_A, \chi_B$ take values in $\left\{{0, 1}\right\}$.
+By definition of $\chi_A$ and $\chi_B$, this is equivalent to the statement that both $s \in A$ and $s \in B$.
+That is, $s \in A \cap B$, by definition of [[Definition:Set Intersection|set intersection]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Tangent Space is Vector Space}
+Tags: 
+
+\begin{theorem}
+Let $M$ be a [[Definition:Smooth Manifold|smooth manifold]] of [[Definition:Dimension of Locally Euclidean Space|dimension]] $n \in \N$. 
+Let $m \in M$ be a [[Definition:Point|point]]. 
+Let $\left({U, \kappa}\right)$ be a [[Definition:Chart|chart]] with $m \in U$. 
+Let $T_m M$ be the [[Definition:Tangent Space|tangent space]] at $m$. 
+Then $T_m M$ is a [[Definition:Real Vector Space|real vector space]] of [[Definition:Dimension of Vector Space|dimension]] $n$, [[Definition:Linear Span|spanned]] by the [[Definition:Basis (Linear Algebra)|basis]]: 
+:$\left\{ {\left.{\dfrac \partial {\partial \kappa^i} }\right\vert_m : i \in \left\{{1, \dotsc, n}\right\} }\right\}$
+that is, the [[Definition:Set|set]] of [[Definition:Partial Derivative|partial derivatives]] with respect to the $i$th [[Definition:Coordinate Function|coordinate function]] $\kappa^i$ evaluated at $m$.
+\end{theorem}
+
+\begin{proof}
+Let $V$ be an [[Definition:Open Neighborhood|open neighborhood]] of $m$ with $V \subseteq U \subseteq M$. 
+Let $C^\infty \left({V, \R}\right)$ be the [[Definition:Set|set]] of [[Definition:Smooth Mapping|smooth mappings]] $f: V \to \R$. 
+Let $X_m, Y_m \in T_m M$.
+Let $\lambda \in \R$.
+Then, by definition of [[Definition:Tangent Vector|tangent vector]] and [[Equivalence of Definitions of Tangent Vector]]:
+:$X_m, Y_m$ are [[Definition:Linear Transformation|linear transformations]] on $C^\infty \left({V, \R}\right)$.  
+Hence $\left({X_m + \lambda Y_m}\right)$ are also [[Definition:Linear Transformation|linear transformations]]. 
+Therefore, it is enough to show that $X_m + \lambda Y_m$ satisfies the [[Definition:Leibniz Law|Leibniz law]]. 
+Let $f, g \in C^\infty \left({V, \R}\right)$. 
+Then:
+{{begin-eqn}}
+{{eqn | l = \left({X_m + \lambda Y_m}\right) \left({f g}\right) 
+      | r = X_m \left({f g}\right) + \lambda Y_m \left({f g}\right)
+      | c = {{Defof|Linear Transformation}}
+}}
+{{eqn | r = X_m \left({f}\right) g \left({m}\right) + f \left({m}\right) X_m \left({g}\right) + \lambda \left({Y_m \left({f}\right) g \left({m}\right) + f \left({m}\right) Y_m \left({g}\right)}\right)
+      | c = [[Definition:Leibniz Law|Leibniz law]] for $X_m, Y_m$ 
+}}
+{{eqn | r = \left({X_m + \lambda Y_m} \right) \left({f}\right) g \left({m}\right) + f \left({m}\right) \left({X_m + \lambda Y_m}\right) \left({g}\right)
+      | c = reordering [[Definition:Summand|summands]]
+}}
+{{end-eqn}}
+It follows that:
+:$X_m + \lambda Y_m \in T_m M$
+Hence $T_m M$ is a [[Definition:Real Vector Space|real vector space]]. 
+Again, by definition of [[Definition:Tangent Vector|tangent vector]] and [[Equivalence of Definitions of Tangent Vector]]:
+:for all $X_m \in T_m M$ there exists a [[Definition:Smooth Curve|smooth curve]]:
+::$\gamma : I \subseteq \R \to M$
+:with $\gamma \left({0}\right) = m$ such that:
+{{begin-eqn}}
+{{eqn | l = X_m \left({f}\right)
+      | r = \left. {\frac {\mathrm d \left({f \circ \gamma} \right)} {\mathrm d \tau} \left({\tau}\right) }\right\rvert_{\tau \mathop = 0}
+}}
+{{eqn | r = \left. {\frac {\mathrm d \left({f \circ \kappa^{-1} \circ \kappa \circ \gamma} \right)} {\mathrm d \tau} \left({\tau}\right)} \right\rvert_{\tau \mathop = 0}
+      | c = $f \circ \kappa^{-1} \circ \kappa = f$, as $\kappa$ is a [[Definition:Homeomorphism|homeomorphism]], in particular a [[Definition:Bijection|bijection]].
+}}
+{{eqn | r = \sum_{i \mathop = 1}^n \left. {\frac {\partial \left({f \circ \kappa^{-1} }\right)} {\partial \kappa^i} \left({\kappa \circ \gamma \left({\tau}\right)}\right) \frac {\mathrm d \left({\kappa^i \circ \gamma}\right)} {\mathrm d \tau}  \left({\tau}\right)} \right\rvert_{\tau \mathop = 0}
+      | c = [[Chain Rule for Real-Valued Functions]]
+}}
+{{eqn | r = \sum_{i \mathop = 1}^n \frac {\mathrm d \left({\kappa^i \circ \gamma} \right)} {\mathrm d \tau} \left({0}\right) \frac {\partial \left({f \circ \kappa^{-1} }\right)} {\partial \kappa^i} \left({\kappa \circ \gamma \left({0}\right)}\right)
+      | c = rearranging
+}}
+{{eqn | r = \sum_{i \mathop = 1}^n \frac {\mathrm d \left({\kappa^i \circ \gamma}\right)} {\mathrm d \tau} \left({0}\right) \frac {\partial \left({f \circ \kappa^{-1} }\right)} {\partial \kappa^i} \left({\kappa \left({m}\right)}\right) 
+      | c = as $m = \gamma \left({0}\right)$
+}}
+{{end-eqn}}
+We define:
+:$X^i_m := \dfrac {\mathrm d \left({\kappa^i \circ \gamma}\right)} {\mathrm d \tau} \left({0}\right)$
+and as above: 
+:$\left.{\dfrac \partial {\partial \kappa^i} }\right\vert_m \left({f}\right) := \dfrac {\partial \left({f \circ \kappa^{-1}  }\right)} {\partial \kappa^i} \left({\kappa \left({m}\right)}\right)$
+Therefore: 
+:$\displaystyle X_m \left({f}\right) = \left({\sum_{i \mathop = 1}^n X^i_m \left.{\dfrac \partial {\partial \kappa^i} }\right\vert_m}\right) \left({f}\right)$
+{{iff}}:
+:$\displaystyle X_m = \sum_{i \mathop = 1}^n X^i_m \left.{\frac \partial {\partial \kappa^i} }\right\vert_m$
+Hence:
+:$\left\{ {\left.{\dfrac \partial {\partial \kappa^i} }\right\vert_m: i \in \left\{ {1, \dotsc, n}\right\} }\right\}$
+forms a [[Definition:Basis (Linear Algebra)|basis]]. 
+Hence, by [[Definition:Dimension of Vector Space]]:
+:$\dim T_m M = n = \dim M$
+This completes the proof. 
+{{qed}}
+o9914y9rjsr1fylrsa9ucf849sw6a91
+\end{proof}<|endoftext|>
+\section{Multiplication of Real Numbers is Right Distributive over Subtraction}
+Tags: Real Multiplication, Real Subtraction, Distributive Operations, Multiplication of Real Numbers Distributes over Subtraction
+
+\begin{theorem}
+{{:Euclid:Proposition/V/6}}
+That is, for any [[Definition:Number|number]] $a$ and for any [[Definition:Integer|integers]] $m, n$:
+:$m a - n a = \paren {m - n} a$
+\end{theorem}
+
+\begin{proof}
+Let two [[Definition:Strictly Positive Real Number|magnitudes]] $AB, CD$ be [[Definition:Equimultiples|equimultiples]] of two magnitudes $E, F$.
+Let $AG, CH$ subtracted from them be [[Definition:Equimultiples|equimultiples]] of the same two $E, F$.
+We need to show that the remainders $GB, HD$ are either equal to $E, F$ or are [[Definition:Equimultiples|equimultiples]] of them.
+:[[File:Euclid-V-6.png|350px]]
+First let $GB = E$.
+Let $CK$ be made equal to $F$.
+We have that $AG$ is the same multiple of $E$ that $CH$ is of $F$, while $GB = E$ and $KC = F$.
+Therefore from [[Multiplication of Numbers Distributes over Addition]], $AB$ is the same multiple of $E$ that $KH$ is of $F$.
+But [[Definition:By Hypothesis|by hypothesis]] $AB$ is the same multiple of $E$ that $CD$ is of $F$.
+Since then, each of the [[Definition:Strictly Positive Real Number|magnitudes]] $KH, CD$ is the same multiple of $F$.
+Therefore $KH = CD$.
+Let $CH$ be subtracted from each.
+Therefore the remainder $KC$ equals the remainder $HD$.
+But $F = KC$, so $HD = F$.
+Hence, if $GB = E$ then $HD = F$.
+Similarly we can prove that, even if $GB$ is a multiple of $E$, then $HD$ is also the same multiple of $F$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Sum of Square Roots as Square Root of Sum}
+Tags: Algebra
+
+\begin{theorem}
+:$\sqrt a + \sqrt b = \sqrt {a + b + \sqrt {4 a b} }$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \sqrt a + \sqrt b
+      | r = \sqrt {\paren {\sqrt a + \sqrt b}^2}
+}}
+{{eqn | r = \sqrt {\sqrt a^2 + \sqrt b^2 + 2 \sqrt a \sqrt b}
+      | c = [[Square of Sum]]
+}}
+{{eqn | r = \sqrt {a + b + \sqrt {4 a b} }
+      | c = [[Exponent Combination Laws/Power of Product|Exponent Combination Laws: Power of Product]]
+}}
+{{end-eqn}}
+{{qed}}
+[[Category:Algebra]]
+dqyudpoaz21as5p0ufny7ers4imbydf
+\end{proof}<|endoftext|>
+\section{Divisor of One of Coprime Numbers is Coprime to Other}
+Tags: Coprime Integers, Divisor of One of Coprime Numbers is Coprime to Other
+
+\begin{theorem}
+Let $a, b \in \N$ be [[Definition:Natural Number|numbers]] such that $a$ and $b$ are [[Definition:Coprime Integers|coprime]]:
+:$a \perp b$
+Let $c > 1$ be a [[Definition:Divisor of Integer|divisor]] of $a$:
+:$c \divides a$
+Then $c$ and $b$ are coprime:
+:$c \perp b$
+{{:Euclid:Proposition/VII/23}}
+\end{theorem}
+
+\begin{proof}
+Let $a \perp b$ and $c > 1: c \divides a$.
+{{AimForCont}} $c \not \perp b$.
+So by definition of [[Definition:Coprime Integers|not coprime]]:
+:$\exists d > 1: d \divides c, d \divides b$
+But from [[Divisor Relation is Transitive]]:
+:$d \divides c, c \divides a \implies d \divides a$
+So $d$ is a [[Definition:Common Divisor of Integers|common divisor]] of both $a$ and $b$.
+So, by definition, $a$ and $b$ are not [[Definition:Coprime Integers|coprime]].
+The result follows by [[Proof by Contradiction]].
+{{qed}}
+\end{proof}
+
+\begin{proof}
+Let $A, B$ be two [[Definition:Natural Number|numbers]] which are [[Definition:Coprime Integers|prime to one another]].
+Let $C$ be any [[Definition:Natural Number|number]] greater than $1$ which [[Definition:Divisor of Integer|measures]] $A$.
+:[[File:Euclid-VII-23.png|250px]]
+Suppose $C$ and $B$ are not [[Definition:Coprime Integers|prime to one another]].
+Then some [[Definition:Natural Number|number]] $D$ will [[Definition:Divisor of Integer|measure]] them both.
+We have that $D$ [[Definition:Divisor of Integer|measures]] $C$ and $C$ [[Definition:Divisor of Integer|measures]] $A$.
+So $D$ [[Definition:Divisor of Integer|measures]] $A$.
+But $D$ also [[Definition:Divisor of Integer|measures]] $B$.
+So $D$ [[Definition:Divisor of Integer|measures]] $A$ and $B$ which are [[Definition:Coprime Integers|prime to one another]].
+By {{EuclidDefLink|VII|12|Relatively Prime}}, this is a [[Proof by Contradiction|contradiction]].
+Therefore there can be no such $D$ that [[Definition:Divisor of Integer|measures]] both $B$ and $C$.
+That is, $B$ and $C$ are [[Definition:Coprime Integers|prime to one another]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Integer Coprime to all Factors is Coprime to Whole/Proof 1}
+Tags: Coprime Integers, Integer Coprime to all Factors is Coprime to Whole
+
+\begin{theorem}
+Let $a, b \in \Z$.
+Let $\displaystyle b = \prod_{j \mathop = 1}^r b_j$
+Let $a$ be [[Definition:Coprime Integers|coprime]] to each of $b_1, \ldots, b_r$.
+Then $a$ is [[Definition:Coprime Integers|coprime]] to $b$.
+\end{theorem}
+
+\begin{proof}
+From [[Integer Combination of Coprime Integers]]:
+:$\forall j \in \set {1, 2, \ldots, r}: a x_j + b_j y_j = 1$
+for some $x_j, y_j \in \Z$.
+Thus:
+:$\displaystyle \prod_{j \mathop = 1}^r b_j y_j = \prod_{j \mathop = 1}^r \paren {1 - a x_j}$
+But $\displaystyle \prod_{j \mathop = 1}^r \paren {1 - a x_j}$ is of the form $1 - a z$.
+Thus:
+{{begin-eqn}}
+{{eqn | l = \prod_{j \mathop = 1}^r b_j \prod_{j \mathop = 1}^r y_j
+      | r = 1 - a z
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = a z + b y
+      | r = 1
+      | c = where $\displaystyle y = \prod_{j \mathop = 1}^r y_j$
+}}
+{{end-eqn}}
+and so, by [[Integer Combination of Coprime Integers]], $a$ and $b$ are [[Definition:Coprime Integers|coprime]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Integer Coprime to all Factors is Coprime to Whole/Proof 2}
+Tags: Integer Coprime to all Factors is Coprime to Whole
+
+\begin{theorem}
+Let $a, b, c \in \Z$.
+Let $c$ be [[Definition:Coprime Integers|coprime]] to each of $a$ and $b$.
+Then $c$ is [[Definition:Coprime Integers|coprime]] to $a b$.
+{{:Euclid:Proposition/VII/24}}
+\end{theorem}
+
+\begin{proof}
+''This proof follows the structure of {{AuthorRef|Euclid}}'s proof, if not its detail.''
+Let $a, b, c \in \Z$ such that $c$ is [[Definition:Coprime Integers|coprime]] to each of $a$ and $b$.
+Let $d = a b$.
+{{AimForCont}} $c$ and $d$ are not [[Definition:Coprime Integers|coprime]].
+Then:
+:$\exists e \in \Z_{>1}: e \divides c, e \divides d$
+We have that $c \perp a$ and $e \divides c$
+From {{EuclidPropLink|book=VII|prop=23|title=Divisor of One of Coprime Numbers is Coprime to Other}}:
+:$e \perp a$
+As $e \divides d$ it follows by definition of [[Definition:Divisor of Integer|divisibility]] that:
+:$\exists f \in \Z: e f = d$
+We also have that:
+:$a b = d$
+So:
+:$a b = e f$
+But from {{EuclidPropLink|book=VII|prop=19|title=Relation of Ratios to Products}}:
+:$e : a = b : f$
+or in more contemporary language:
+:$\dfrac a e = \dfrac b f$
+But $a \perp e$.
+From {{EuclidPropLink|book = VII|prop = 21|title = Coprime Numbers form Fraction in Lowest Terms}}, $\dfrac a e$ is in [[Definition:Canonical Form of Rational Number|canonical form]].
+From {{EuclidPropLink|book = VII|prop = 20|title = Ratios of Fractions in Lowest Terms}}:
+:$e \divides b$
+But we already have that:
+:$e \divides c$
+So $e$ is a [[Definition:Divisor of Integer|divisor]] of both $b$ and $c$.
+But this is a [[Proof by Contradiction|contradiction]] of the assertion that $c$ and $b$ are [[Definition:Coprime Integers|coprime]].
+Hence the result.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Square of Coprime Number is Coprime}
+Tags: Coprime Integers
+
+\begin{theorem}
+Let $a$ and $b$ be [[Definition:Coprime Integers|coprime integers]]:
+:$a, b \in \Z: a \perp b$
+Then:
+:$a^2 \perp b$
+{{:Euclid:Proposition/VII/25}}
+\end{theorem}
+
+\begin{proof}
+Let $a \perp b$.
+Let $a^2 = c$.
+Let $d = a$.
+As $a \perp b$ it follows that $d \perp b$.
+From {{EuclidPropLink|book=VII|prop=24|title=Integer Coprime to all Factors is Coprime to Whole}}:
+: $a d \perp b$
+But $a d = c = a^2$.
+Hence the result.
+{{qed}}
+{{Euclid Note|25|VII}}
+\end{proof}<|endoftext|>
+\section{Product of Coprime Pairs is Coprime}
+Tags: Coprime Integers
+
+\begin{theorem}
+Let $a, b, c, d$ be [[Definition:Integer|integers]].
+Let: 
+:$a \perp c, b \perp c, a \perp d, b \perp d$
+where $a \perp c$ denotes that $a$ and $c$ are [[Definition:Coprime Integers|coprime]].
+Then:
+:$a b \perp c d$
+{{:Euclid:Proposition/VII/26}}
+\end{theorem}
+
+\begin{proof}
+Let $e = a b, f = c d$.
+{{begin-eqn}}
+{{eqn | l = a
+      | o = \perp
+      | r = c
+      | c = 
+}}
+{{eqn | ll= \land
+      | l = b
+      | o = \perp
+      | r = c
+      | c = 
+}}
+{{eqn | n = 1
+      | lll= \implies
+      | l = a b
+      | o = \perp
+      | r = c
+      | c = {{EuclidPropLink|book=VII|prop=24|title=Integer Coprime to all Factors is Coprime to Whole}}
+}}
+{{eqn | l = a
+      | o = \perp
+      | r = d
+      | c = 
+}}
+{{eqn | ll= \land
+      | l = b
+      | o = \perp
+      | r = d
+      | c = 
+}}
+{{eqn | n = 2
+      | lll= \implies
+      | l = a b
+      | o = \perp
+      | r = d
+      | c = {{EuclidPropLink|book=VII|prop=24|title=Integer Coprime to all Factors is Coprime to Whole}}
+}}
+{{eqn | l = a b
+      | o = \perp
+      | r = c
+      | c = from $(1)$
+}}
+{{eqn | ll= \land
+      | l = a b
+      | o = \perp
+      | r = d
+      | c = from $(2)$
+}}
+{{eqn | lll= \implies
+      | l = a b
+      | o = \perp
+      | r = c d
+      | c = {{EuclidPropLink|book=VII|prop=24|title=Integer Coprime to all Factors is Coprime to Whole}}
+}}
+{{end-eqn}}
+{{qed}}
+{{Euclid Note|26|VII}}
+\end{proof}<|endoftext|>
+\section{Powers of Coprime Numbers are Coprime}
+Tags: Coprime Integers
+
+\begin{theorem}
+Let $a, b$ be [[Definition:Coprime Integers|coprime integers]]:
+:$a \perp b$
+Then:
+:$\forall n \in \N_{>0}: a^n \perp b^n$
+{{:Euclid:Proposition/VII/27}}
+\end{theorem}
+
+\begin{proof}
+Proof by [[Principle of Mathematical Induction|induction]]:
+Let  $a \perp b$.
+For all $n \in \N_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
+:$a^n \perp b^n$
+$\map P 1$ is true, as this just says:
+:$a \perp b$
+=== Basis for the Induction ===
+By {{EuclidPropLink|book = VII|prop = 25|title = Square of Coprime Number is Coprime}}:
+:$a^2 \perp b$
+Again, by {{EuclidPropLink|book = VII|prop = 25|title = Square of Coprime Number is Coprime}}:
+:$a^2 \perp b^2$
+This is our [[Definition:Basis for the Induction|basis for the induction]].
+=== Induction Hypothesis ===
+Now we need to show that, if $\map P k$ is true, where $k \ge 2$, then it logically follows that $\map P {k + 1}$ is true.
+So this is our [[Definition:Induction Hypothesis|induction hypothesis]]:
+:$a^k \perp b^k$
+Then we need to show:
+:$a^{k + 1} \perp b^{k + 1}$
+=== Induction Step ===
+This is our [[Definition:Induction Step|induction step]]:
+{{begin-eqn}}
+{{eqn | l = a^k 
+      | o = \perp
+      | r = b^k
+      | c = 
+}}
+{{eqn | ll=\leadsto
+      | l = a^k \times a
+      | o = \perp
+      | r = b^k
+      | c = {{EuclidPropLink|book = VII|prop = 24|title = Integer Coprime to all Factors is Coprime to Whole}}
+}}
+{{eqn | ll=\leadsto
+      | l = a^{k + 1}
+      | o = \perp
+      | r = b^k
+      | c = 
+}}
+{{eqn | ll=\leadsto
+      | l = a^{k + 1}
+      | o = \perp
+      | r = b^k \times b
+      | c = {{EuclidPropLink|book = VII|prop = 24|title = Integer Coprime to all Factors is Coprime to Whole}}
+}}
+{{eqn | ll=\leadsto
+      | l = a^{k + 1}
+      | o = \perp
+      | r = b^{k + 1}
+      | c = 
+}}
+{{end-eqn}}
+So $\map P k \implies \map P {k + 1}$ and the result follows by the [[Principle of Mathematical Induction]].
+Therefore:
+:$\forall n \in \N_{>0}: a^n \perp b^n$
+{{qed}}
+{{Euclid Note|27|VII|Euclid's proof does not use the full induction process, which is a later mathematical innovation.}}
+\end{proof}<|endoftext|>
+\section{Numbers are Coprime iff Sum is Coprime to Both}
+Tags: Coprime Integers
+
+\begin{theorem}
+Let $a, b$ be [[Definition:Integer|integers]].
+Then:
+:$a \perp b \iff a \perp \paren {a + b}$
+where $a \perp b$ denotes that $a$ and $b$ are [[Definition:Coprime Integers|coprime]].
+{{:Euclid:Proposition/VII/28}}
+\end{theorem}
+
+\begin{proof}
+=== Necessary Condition ===
+Let $a \perp b$.
+Suppose $a + b$ is not [[Definition:Coprime Integers|coprime]] to $a$.
+Then:
+:$\exists d \in \Z_{>1}: d \divides a, d \divides \paren {a + b}$
+But then:
+:$d \divides \paren {\paren {a + b} - a}$
+and so:
+:$d \divides b$
+and so $a$ and $b$ are not [[Definition:Coprime Integers|coprime]].
+From this [[Proof by Contradiction|contradiction]] it follows that $a + b$ is [[Definition:Coprime Integers|coprime]] to $a$.
+{{qed|lemma}}
+=== Sufficient Condition ===
+Let $a + b$ be [[Definition:Coprime Integers|coprime]] to $a$.
+Suppose $a$ is not [[Definition:Coprime Integers|coprime]] to $b$.
+Then:
+:$\exists d \in \Z_{>1}: d \divides a, d \divides b$
+and so:
+:$d \divides \paren {a + b}$
+and so $a$ and $\paren {a + b}$ are not [[Definition:Coprime Integers|coprime]].
+From this [[Proof by Contradiction|contradiction]] it follows that $a$ is [[Definition:Coprime Integers|coprime]] to $b$.
+{{qed}}
+{{Euclid Note|28|VII}}
+\end{proof}<|endoftext|>
+\section{Composite Number has Prime Factor}
+Tags: Prime Numbers
+
+\begin{theorem}
+Let $a$ be a [[Definition:Composite Number|composite number]].
+Then there exists a [[Definition:Prime Number|prime number]] $p$ such that:
+:$p \divides a$
+where $\divides$ means '''is a [[Definition:Divisor of Integer|divisor]] of'''.
+{{:Euclid:Proposition/VII/31}}
+\end{theorem}
+
+\begin{proof}
+By definition of [[Definition:Composite Number|composite number]]:
+:$\exists b \in \Z: b \divides a$
+If $b$ is a [[Definition:Prime Number|prime number]] then the proof is complete.
+Otherwise $b$ is [[Definition:Composite Number|composite]].
+By definition of [[Definition:Composite Number|composite number]]:
+:$\exists c \in \Z: c \divides b$
+and so:
+:$c \divides a$
+Again, if $c$ is a [[Definition:Prime Number|prime number]] then the proof is complete.
+Continuing in this manner, some [[Definition:Prime Number|prime number]] will be found which will [[Definition:Divisor of Integer|divide]] the [[Definition:Natural Number|number]] before it.
+This will be a [[Definition:Divisor of Integer|divisor]] of $a$.
+Because if not, then an [[Definition:Infinite Sequence|infinite series]] of [[Definition:Natural Number|numbers]] will be [[Definition:Divisor of Integer|divisors]] of $a$, each of which is less than the other.
+This is impossible in [[Definition:Natural Number|numbers]].
+Hence the result.
+{{qed}}
+{{Euclid Note|31|VII|}}
+\end{proof}<|endoftext|>
+\section{Natural Number is Prime or has Prime Factor}
+Tags: Prime Numbers
+
+\begin{theorem}
+Let $a$ be a [[Definition:Natural Number|natural number]] greater than $1$.
+Then either:
+:$a$ is a [[Definition:Prime Number|prime number]]
+or:
+:there exists a [[Definition:Prime Number|prime number]] $p \ne a$ such that $p \divides a$
+where $\divides$ denotes '''is a [[Definition:Divisor of Integer|divisor]] of'''.
+{{:Euclid:Proposition/VII/32}}
+\end{theorem}
+
+\begin{proof}
+By definition of [[Definition:Composite Number|composite number]] $a$ is either [[Definition:Prime Number|prime]] or [[Definition:Composite Number|composite]].
+Let $a$ be [[Definition:Prime Number|prime]].
+Then the statement of the result is fulfilled.
+Let $a$ be [[Definition:Composite Number|composite]].
+Then by {{EuclidPropLink|book = VII|prop = 31|title = Composite Number has Prime Factor}}:
+:$\exists p: p \divides a$
+where $p$ is a [[Definition:Prime Number|prime number]].
+The result follows by [[Proof by Cases]].
+{{qed}}
+{{Euclid Note|32|VII|}}
+\end{proof}<|endoftext|>
+\section{LCM of Three Numbers}
+Tags: Lowest Common Multiple
+
+\begin{theorem}
+Let $a, b, c \in \Z: a b c \ne 0$.
+The [[Definition:Lowest Common Multiple of Integers|lowest common multiple]] of $a, b, c$, denoted $\lcm \set {a, b, c}$, can always be found.
+{{:Euclid:Proposition/VII/36}}
+\end{theorem}
+
+\begin{proof}
+Let $d = \lcm \set {a, b}$.
+This exists from {{EuclidPropLink|book = VII|prop = 34|title = Existence of Lowest Common Multiple}}.
+Either $c \divides d$ or not, where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
+Suppose $c \divides d$.
+But by definition of [[Definition:Lowest Common Multiple of Integers|lowest common multiple]], $a \divides d$ and $b \divides d$ also.
+Suppose $a, b, c$ are [[Definition:Divisor of Integer|divisors]] of $e$ where $e < d$.
+Then $a, b$ are [[Definition:Divisor of Integer|divisors]] of $e$.
+That is, $e$ is a [[Definition:Common Divisor of Integers|common divisor]] of $a$ and $b$ which is lower than $d$.
+But by {{EuclidPropLink|book = VII|prop = 35|title = LCM Divides Common Multiple}}:
+:$d \divides e$
+which is impossible.
+It follows that there can be no such $e$.
+Therefore $d = \lcm \set {a, b, c}$.
+Now suppose $c \nmid d$.
+Let $e = \lcm \set {c, d}$.
+This exists from {{EuclidPropLink|book = VII|prop = 34|title = Existence of Lowest Common Multiple}}.
+Since $a$ and $b$ are both [[Definition:Divisor of Integer|divisors]] of $d$, it follows that:
+:$a \divides e$
+:$b \divides e$
+But we have that $c \divides e$ as well.
+Suppose $a, b, c$ are [[Definition:Divisor of Integer|divisors]] of $f$ where $f < e$.
+Then $a, b$ are [[Definition:Divisor of Integer|divisors]] of $f$.
+So by {{EuclidPropLink|book = VII|prop = 35|title = LCM Divides Common Multiple}}:
+:$d = \lcm \set {a, b} \divides f$
+But also $c \divides f$.
+Therefore $c$ and $d$ are [[Definition:Divisor of Integer|divisors]] of $f$.
+By {{EuclidPropLink|book = VII|prop = 35|title = LCM Divides Common Multiple}}:
+:$e = \lcm \set {c, d} \divides f$
+But this is impossible as [[Definition:By Hypothesis|by hypothesis]] $f < e$.
+Therefore $a, b, c$ are [[Definition:Divisor of Integer|divisors]] of no number smaller than $e$.
+Therefore $e = \lcm \set {a, b, c}$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Negative of Real Function that Increases Without Bound}
+Tags: Unbounded Mappings
+
+\begin{theorem}
+Let $f: \R \to \R$ be a [[Definition:Real Function|real function]].
+Then:
+:$(1): \quad \displaystyle \lim_{x \mathop \to +\infty} \map f x = +\infty \implies \lim_{x \mathop \to +\infty} -\map f x = -\infty$
+:$(2): \quad \displaystyle \lim_{x \mathop \to -\infty} \map f x = +\infty \implies \lim_{x \mathop \to -\infty} -\map f x = -\infty$
+\end{theorem}
+
+\begin{proof}
+Suppose $\displaystyle  \lim_{x \mathop \to +\infty} \map f x = +\infty$.
+Then by the definition of [[Definition:Infinite Limit at Infinity|infinite limits at infinity]]:
+:$\forall M > 0: \exists N > 0: x > N \implies \map f x > M$
+But $M > 0 \iff -M < 0$.
+Likewise $\map f x > M \iff -\map f x < -M$.
+Putting $M' = -M$:
+:$\forall M' < 0: \exists N > 0: x > N \implies -\map f x < M'$
+The result then follows from the definition of [[Definition:Negative Infinite Limit at Infinity|negative infinite limit at infinity]].
+The proof for $\displaystyle \lim_{x \mathop \to -\infty} \map f x = +\infty$ is analagous.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Integer Divided by Divisor is Integer}
+Tags: Divisors
+
+\begin{theorem}
+Let $a, b \in \N$.
+Then:
+:$b \divides a \implies \dfrac 1 b \times a \in \N$
+where $\divides$ denotes [[Definition:Divisor of Integer|divisibilty]].
+{{:Euclid:Proposition/VII/37}}
+\end{theorem}
+
+\begin{proof}
+Let $b \divides a$.
+By definition of [[Definition:Divisor of Integer|divisibilty]]:
+:$\exists c \in \N: c \times b = a$
+Then also:
+:$c \times 1 = c$
+So by {{EuclidPropLink|book = VII|prop = 15|title = Alternate Ratios of Multiples}}:
+:$1 : b = c : a$
+Hence the result.
+{{qed}}
+{{Euclid Note|37|VII|}}
+\end{proof}<|endoftext|>
+\section{Divisor is Reciprocal of Divisor of Integer}
+Tags: Divisors
+
+\begin{theorem}
+Let $a, b, c \in \Z_{>0}$.
+Then:
+:$b = \dfrac 1 c \times a \implies c \divides a$
+where $\divides$ denotes [[Definition:Divisor of Integer|divisibilty]].
+{{:Euclid:Proposition/VII/38}}
+\end{theorem}
+
+\begin{proof}
+Let $a$ have an [[Definition:Aliquot Part|aliquot part]] $b$.
+Let $c$ be an [[Definition:Integer|integer]] called by the same name as the [[Definition:Aliquot Part|aliquot part]] $b$.
+Then:
+:$1 = \dfrac 1 c \times c$
+and so by {{EuclidPropLink|book = VII|prop = 15|title = Alternate Ratios of Multiples}}:
+:$ 1 : c = b : a$
+Hence the result.
+{{qed}}
+{{Euclid Note|38|VII|}}
+\end{proof}<|endoftext|>
+\section{Negative of Real Function that Decreases Without Bound}
+Tags: Unbounded Mappings
+
+\begin{theorem}
+Let $f: \R \to \R$ be a [[Definition:Real Function|real function]].
+Then:
+:$(1): \quad \displaystyle \lim_{x \mathop \to +\infty} \map f x = -\infty \implies \lim_{x \mathop \to +\infty} -\map f x = +\infty$
+:$(2): \quad \displaystyle \lim_{x \mathop \to -\infty} \map f x = -\infty \implies \lim_{x \mathop \to -\infty} -\map f x = +\infty$
+\end{theorem}
+
+\begin{proof}
+Suppose $\displaystyle \lim_{x \mathop \to +\infty} \map f x = -\infty$.
+Then by the definition of [[Definition:Negative Infinite Limit at Infinity|negative infinite limit at infinity]]:
+:$\forall M < 0: \exists N > 0: x > N \implies \map f x < M$
+But $M < 0 \iff -M > 0$.
+Likewise $\map f x < M \iff -\map f x > -M$.
+Putting $M' = -M$:
+:$\forall M' > 0: \exists N > 0: x > N \implies -\map f x > M'$
+The result then follows from the definition of [[Definition:Infinite Limit at Infinity|infinite limit at infinity]].
+The proof for $\displaystyle \lim_{x \mathop \to -\infty} \map f x = -\infty$ is analogous.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Construction of Geometric Sequence in Lowest Terms}
+Tags: Geometric Sequences of Integers, Construction of Geometric Sequence in Lowest Terms
+
+\begin{theorem}
+It is possible to find a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]] $G_n$ of [[Definition:Length of Sequence|length]] $n + 1$ with a given [[Definition:Common Ratio of Geometric Sequence|common ratio]] such that $G_n$ is in its [[Definition:Geometric Sequence of Integers in Lowest Terms|lowest terms]].
+{{:Euclid:Proposition/VIII/2}}
+\end{theorem}
+
+\begin{proof}
+Let $r = \dfrac a b$ be the given [[Definition:Common Ratio of Geometric Sequence|common ratio]].
+Let the required [[Definition:Geometric Sequence|geometric sequence]] have a [[Definition:Length of Sequence|length]] of $4$.
+Let $a^2 = c$.
+Let $a b = d$.
+Let $b^2 = e$.
+Let:
+:$a c = f$
+:$a d = g$
+:$a e = h$
+and let:
+:$b e = k$
+As:
+:$a^2 = c$
+:$a b = d$
+it follows from {{EuclidPropLink|book = VII|prop = 17|title = Multiples of Ratios of Numbers}} that:
+:$\dfrac a b = \dfrac c d$
+As:
+:$a b = d$
+:$b^2 = e$
+it follows from {{EuclidPropLink|book = VII|prop = 18|title = Ratios of Multiples of Numbers}} that:
+:$\dfrac a b = \dfrac d e$
+As:
+:$a c = f$
+:$a d = g$
+it follows from {{EuclidPropLink|book = VII|prop = 17|title = Multiples of Ratios of Numbers}} that:
+:$\dfrac c d = \dfrac f g$
+As:
+:$a d = g$
+:$a e = h$
+it follows from {{EuclidPropLink|book = VII|prop = 17|title = Multiples of Ratios of Numbers}} that:
+:$\dfrac d e = \dfrac g h$
+As:
+:$a e = h$
+:$b e = k$
+it follows from {{EuclidPropLink|book = VII|prop = 18|title = Ratios of Multiples of Numbers}} that:
+:$\dfrac a b = \dfrac h k$
+Putting the above together:
+:$c, d, e$ are in [[Definition:Geometric Sequence|geometric sequence]] with [[Definition:Common Ratio of Geometric Sequence|common ratio]] $\dfrac a b$
+:$f, g, h, k$ are in [[Definition:Geometric Sequence|geometric sequence]] with [[Definition:Common Ratio of Geometric Sequence|common ratio]] $\dfrac a b$
+We have that $a$ and $b$ are the smallest numbers with the same [[Definition:Ratio|ratio]].
+So by {{EuclidPropLink|book = VII|prop = 22|title = Numbers forming Fraction in Lowest Terms are Coprime}}:
+:$a \perp b$
+where $\perp$ denotes [[Definition:Coprime Integers|coprimality]].
+We also have that:
+:$a^2 = c, b^2 = e$
+and:
+:$a c = e, b e = k$
+so by {{EuclidPropLink|book = VII|prop = 27|title = Powers of Coprime Numbers are Coprime}}:
+:$c \perp e$
+:$f \perp k$
+But from {{EuclidPropLink|book = VIII|prop = 1|title = Geometric Sequence with Coprime Extremes is in Lowest Terms}}, these are the least of those with the same [[Definition:Common Ratio of Geometric Sequence|common ratio]].
+Hence the result.
+{{qed}}
+\end{proof}
+
+\begin{proof}
+Let the required [[Definition:Length of Sequence|length]] of the [[Definition:Geometric Sequence|geometric sequence]] $P$ be $n$.
+Let $r$ be the given [[Definition:Common Ratio of Geometric Sequence|common ratio]].
+From [[Common Ratio in Integer Geometric Sequence is Rational]], $r$ is a [[Definition:Rational Number|rational number]].
+Let $r = \dfrac p q$ be in [[Definition:Canonical Form of Rational Number|canonical form]].
+Thus, by definition:
+:$p \perp q$
+Let $a$ be the first [[Definition:Term of Geometric Sequence|term]] of $P$.
+Then the [[Definition:Sequence|sequence]] $P$ is:
+:$P = \tuple {a, a \dfrac p q, a \dfrac {p^2} {q^2}, \ldots, a \dfrac {p^n} {q^n} }$
+All the elements of $P$ are [[Definition:Natural Number|natural numbers]], so, in particular:
+:$a \dfrac {p^n} {q^n} \in \N$
+From [[Powers of Coprime Numbers are Coprime]]:
+:$p^n \perp q^n$
+and so from [[Euclid's Lemma]]:
+:$q^n \divides a$
+Thus:
+:$a = k q^n$
+for some $k \in \N$, and so:
+:$P = \tuple {k q^n, k p q^{n - 1}, k p^2 q^{n - 2}, \ldots, k p^{n - 1} q, k p^n}$
+From [[Geometric Sequence with Coprime Extremes is in Lowest Terms]]:
+:$k q^n \perp k p^n$
+from which it follows that $k = 1$.
+It follows that the required [[Definition:Geometric Sequence|geometric sequence]] is:
+:$P = \tuple {q^n, p q^{n - 1}, p^2 q^{n - 2}, \ldots, p^{n - 1} q, p^n}$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Geometric Sequence in Lowest Terms has Coprime Extremes}
+Tags: Geometric Sequences of Integers, Geometric Sequence in Lowest Terms has Coprime Extremes
+
+\begin{theorem}
+A [[Definition:Geometric Sequence of Integers in Lowest Terms|geometric sequence of integers in lowest terms]] has [[Definition:Extremes of Finite Geometric Sequence|extremes]] which are [[Definition:Coprime Integers|coprime]].
+{{:Euclid:Proposition/VIII/3}}
+\end{theorem}
+
+\begin{proof}
+Let $a_0, a_1, a_2, \ldots, a_n$ be [[Definition:Natural Number|natural numbers]].
+Let $\sequence {a_k}_{0 \mathop \le k \mathop \le n}$ be a [[Definition:Geometric Sequence|geometric sequence]] with [[Definition:Common Ratio|common ratio]] $r$.
+Let $a_0, a_1, \ldots, a_n$ be the smallest such [[Definition:Natural Number|natural numbers]].
+From {{EuclidPropLink|book = VII|prop = 33|title = Least Ratio of Numbers}}, let $d_0, d_1$ be the smallest [[Definition:Natural Number|natural numbers]] such that $d_1 = r d_0$.
+From {{EuclidPropLink|book = VIII|prop = 2|title = Construction of Geometric Sequence in Lowest Terms}} one can build a sequence of $3, 4, \ldots, n$ terms with the same property.
+Let the [[Definition:Geometric Sequence|geometric sequence]] so constructed with $n$ terms be $\sequence {b_k}_{0 \mathop \le k \mathop \le n}$.
+From {{EuclidPropLink|book = VII|prop = 22|title = Numbers forming Fraction in Lowest Terms are Coprime}}, $d_0$ and $d_1$ are [[Definition:Coprime Integers|coprime]].
+From {{EuclidPropLink|book = VIII|prop = 2|qual = Porism|title = Construction of Geometric Sequence in Lowest Terms}}, each of the extreme terms of the intermediate [[Definition:Geometric Sequence|geometric sequences]] is a [[Definition:Integer Power|power]] of a [[Definition:Natural Number|natural number]].
+From {{EuclidPropLink|book = VII|prop = 27|title = Powers of Coprime Numbers are Coprime}}, each of those extreme terms is [[Definition:Coprime Integers|coprime]].
+We have that $a_0, \ldots, a_n$ are the smallest such [[Definition:Natural Number|natural numbers]] such that $\sequence {a_k}_{0 \mathop \le k \mathop \le n}$ is a [[Definition:Geometric Sequence|geometric sequence]] with [[Definition:Common Ratio|common ratio]] $r$.
+We have also constructed $b_0, \ldots, b_n$ to have the same property.
+Thus:
+: $a_0 = b_0, a_1 = b_1, \ldots, a_n = b_n$
+and the result follows.
+{{qed}}
+\end{proof}
+
+\begin{proof}
+Let $P$ be a [[Definition:Geometric Sequence|geometric sequence]] of [[Definition:Natural Number|natural numbers]] of [[Definition:Length of Sequence|length]] $n$.
+Let the [[Definition:Common Ratio|common ratio]] of $P$ be expressed in [[Definition:Canonical Form of Rational Number|canonical form]] as $\dfrac p q$.
+From [[Construction of Geometric Sequence in Lowest Terms]]:
+:$P = \paren {q^n, p q^{n - 1}, p^2 q^{n - 2}, \ldots, p^{n - 1} q, p^n}$
+By definition of [[Definition:Canonical Form of Rational Number|canonical form]]:
+:$p \perp q$
+It follows from [[Powers of Coprime Numbers are Coprime]] that:
+:$p^n \perp q^n$
+Hence the result.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{General Associativity Theorem/Formulation 3}
+Tags: General Associativity Theorem
+
+\begin{theorem}
+Let $\struct {S, \circ}$ be a [[Definition:Semigroup|semigroup]].
+Let $a_i$ denote [[Definition:Element|elements]] of $S$.
+Let $\circ$ be [[Definition:Associative|associative]].
+Let $n \in \Z$ be a [[Definition:Positive Integer|positive integer]] such that $n \ge 3$.
+Then all possible [[Definition:Parenthesization|parenthesizations]] of the expression:
+:$a_1 \circ a_2 \circ \cdots \circ a_n$
+are equivalent.
+\end{theorem}
+
+\begin{proof}
+Let $\circ$ be [[Definition:Associative|associative]].
+It will be shown that any [[Definition:Parenthesization|parenthesization]] of $a_1 \circ a_2 \circ \dots \circ a_n$ is equal to the '''left-associated expression''':
+:$\paren {\paren {\paren {\cdots \paren {a_1 \circ a_2} \circ a_3} \circ \cdots} \circ a_n}$
+The proof proceeds by [[Principle of Mathematical Induction|induction]] on $n$.
+=== Basis for the Induction ===
+The base case, $n = 3$:
+The product $a_1 \circ a_2 \circ a_3$ has two [[Definition:Parenthesization|parenthesizations]]:
+:$P_1: a_1 \circ \paren {a_2 \circ a_3}$
+:$P_2: \paren {a_1 \circ a_2} \circ a_3$
+$P_2$ is the left-associative [[Definition:Parenthesization|parenthesization]] for $n = 3$.
+From the associativity condition, $P_1$ and $P_2$ are identical.
+This is the [[Principle of Mathematical Induction#Basis for the Induction|basis for the induction]].
+=== Induction Hypothesis ===
+If for all $m < n$, all [[Definition:Parenthesization|parenthesizations]] of the $m$-product are identical to its left-associated [[Definition:Parenthesization|parenthesization]]:
+:$\paren {\paren {\paren {\cdots \paren {a_1 \circ a_2} \circ a_3} \circ \cdots} \circ a_m}$
+then all [[Definition:Parenthesization|parenthesizations]] of the $n$-product are identical to its left-associated [[Definition:Parenthesization|parenthesization]]:
+:$\paren {\paren {\paren {\cdots \paren {a_1 \circ a_2} \circ a_3} \circ \cdots} \circ a_n}$
+This is the [[Principle of Mathematical Induction#Induction Hypothesis|induction hypothesis]].
+=== Induction Step ===
+Let $P_i$ denote any [[Definition:Parenthesization|parenthesization]] of $a_1 \circ a_2 \circ \dots \circ a_n$.
+Then $P_i$ is always the product of two smaller products:
+{{begin-eqn}}
+{{eqn | l = P_i
+      | r = A \circ B
+      | c = outermost multiplication
+}}
+{{eqn | r = \paren {a_1 \circ a_2 \circ \cdots \circ a_m} \circ \paren {a_{m + 1} \circ a_{m + 2} \circ \cdots \circ a_n}
+      | c =
+}}
+{{eqn | r = \paren {\paren {\paren {\paren {\cdots \paren {a_1 \circ a_2} \circ a_3} \circ \cdots} \circ a_m} \circ \paren {\paren {\paren {\cdots \paren {a_{m + 1} \circ a_{m + 2} } \circ a_{m + 3} } \circ \cdots} \circ a_n} }
+      | c = [[General Associativity Theorem/Formulation 3#Induction Hypothesis|Induction Hypothesis]]
+}}
+{{eqn | r = \paren {\paren {\paren {\cdots \paren {a_1 \circ a_2} \circ a_3} \circ \cdots} \circ a_{n - 1} } \circ a_n
+      | c = {{Defof|Associative Operation}}
+}}
+{{end-eqn}}
+By the [[Principle of Mathematical Induction]], the proof is complete.
+{{qed|lemma}}
+It follows that all [[Definition:Parenthesization|parenthesizations]] of $a_1 \circ a_2 \circ \dots \circ a_n$ yield identical results.
+So the theorem holds.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{First Element of Geometric Sequence not dividing Second}
+Tags: Geometric Sequences of Integers, First Element of Geometric Sequence not dividing Second
+
+\begin{theorem}
+Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]] of [[Definition:Length of Sequence|length]] $n$.
+Let $a_0$ not be a [[Definition:Divisor of Integer|divisor]] of $a_1$.
+Then:
+:$\forall j, k \in \set {0, 1, \ldots, n}, j \ne k: a_j \nmid a_k$
+That is, if the [[Definition:Initial Term of Geometric Sequence|initial term]] of $P$ does not [[Definition:Divisor of Integer|divide]] the second, no [[Definition:Term of Geometric Sequence|term]] of $P$ [[Definition:Divisor of Integer|divides]] any other [[Definition:Term of Geometric Sequence|term]] of $P$.
+{{:Euclid:Proposition/VIII/6}}
+\end{theorem}
+
+\begin{proof}
+Let $P_a = \tuple {a_0, a_1, \ldots, a_n}$ be a [[Definition:Geometric Sequence|geometric sequence]] of [[Definition:Natural Number|natural numbers]] such that $a_0 \nmid a_1$.
+{{AimForCont}} $a_0 \divides a_k$ for some $k: 2 \le k \le n$.
+Let $b_0, b_1, \ldots, b_k$ be the least [[Definition:Natural Number|natural numbers]] which have the same [[Definition:Common Ratio of Geometric Sequence|common ratio]] as $a_0, a_1, \ldots, a_k$.
+These can be found by means of {{EuclidPropLink|book = VII|prop = 33|title = Least Ratio of Numbers}}.
+From {{EuclidPropLink|book = VII|prop = 14|title = Proportion of Numbers is Transitive}}
+:$a_0 : a_k = b_0 : b_k$
+Also:
+:$a_0 : a_1 = b_0 : b_1$
+and so as $a_0 \nmid a_1$ it follows by {{EuclidDefLink|VII|20|Proportional}}:
+:$b_0 \nmid b_1$
+From [[One Divides all Integers]] it follows that:
+:$b_0 \ne 1$
+From {{EuclidPropLink|book = VIII|prop = 3|title = Construction of Sequence of Numbers with Given Ratios}}:
+:$b_0 \perp b_k$
+But as:
+:$a_0 : a_k = b_0 : b_k$
+it follows that:
+:$a_0 \nmid a_k$
+Now suppose $a_j \divides a_k$ such that $0 < j < k$.
+Let $b_j, \ldots, b_k$ be the least [[Definition:Natural Number|natural numbers]] which have the same [[Definition:Common Ratio of Geometric Sequence|common ratio]] as $a_j, \ldots, a_k$.
+These can be found by means of {{EuclidPropLink|book = VII|prop = 33|title = Least Ratio of Numbers}}.
+From {{EuclidPropLink|book = VII|prop = 14|title = Proportion of Numbers is Transitive}}:
+:$a_j : a_k = b_j : b_k$
+The other cases follow similarly.
+{{qed}}
+\end{proof}
+
+\begin{proof}
+From [[Form of Geometric Sequence of Integers]], the [[Definition:Term of Geometric Sequence|terms]] of $P$ are in the form:
+:$\paren 1: \quad: a_j = k q^j p^{n - j}$
+where $p \perp q$.
+Let $a_0 \nmid a_1$.
+That is:
+{{begin-eqn}}
+{{eqn | l = a_0
+      | o = \nmid
+      | r = a_1
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = k q^n
+      | o = \nmid
+      | r = k p q^{n - 1}
+      | c = from $\paren 1$
+}}
+{{eqn | ll= \leadsto
+      | l = \paren {k q^{n - 1} } q
+      | o = \nmid
+      | r = \paren {k q^{n - 1} } p
+      | c = rearranging
+}}
+{{eqn | ll= \leadsto
+      | l = q
+      | o = \nmid
+      | r = p
+      | c = [[Definition:Contrapositive Statement|Contrapositive]] of [[Multiple of Divisor Divides Multiple]]
+}}
+{{end-eqn}}
+Aiming for a [[Proof by Contradiction|contradiction]], suppose that:
+: $\exists i, j \in \set {0, 1, \ldots, n}, i \ne j: a_i \divides a_j$
+{{WLOG}} let $i < j$.
+{{begin-eqn}}
+{{eqn | l = a_i
+      | o = \divides
+      | r = a_j
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = k q^i p^{n - i}
+      | o = \divides
+      | r = k q^j p^{n - j}
+      | c = from $\paren 1$
+}}
+{{eqn | ll= \leadsto
+      | l = m \paren {k q^j p^{n - i} } q^{j - i}
+      | r = \paren {k q^j p^{n - i} } p^{j - i}
+      | c = for some $m \in \Z$
+}}
+{{eqn | ll= \leadsto
+      | l = m q^{j - i}
+      | r = p^{j - i}
+      | c = [[Definition:Contrapositive Statement|Contrapositive]] of [[Multiple of Divisor Divides Multiple]]
+}}
+{{eqn | ll= \leadsto
+      | l = q^{j - i}
+      | o = \divides
+      | r = p^{j - i}
+      | c = 
+}}
+{{end-eqn}}
+But we have that $p \perp q$.
+From [[Powers of Coprime Numbers are Coprime]]:
+:$q^{j - i} \perp p^{j - i}$
+This can happen only when $q = 1$.
+This is incompatible with $q \nmid p$.
+From this contradiction, the result follows.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{First Element of Geometric Sequence that divides Last also divides Second}
+Tags: Geometric Sequences of Integers
+
+\begin{theorem}
+Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]] of [[Definition:Length of Sequence|length]] $n$.
+Let $a_0$ be a [[Definition:Divisor of Integer|divisor]] of $a_n$.
+Then $a_0$ is a [[Definition:Divisor of Integer|divisor]] of $a_2$.
+{{:Euclid:Proposition/VIII/7}}
+\end{theorem}
+
+\begin{proof}
+[[Definition:By Hypothesis|By hypothesis]], let $a_0$ be a [[Definition:Divisor of Integer|divisor]] of $a_n$.
+{{AimForCont}} $a_0$ is not a [[Definition:Divisor of Integer|divisor]] of $a_2$.
+From [[First Element of Geometric Sequence not dividing Second]] it would follow that $a_0$ does not divide $a_n$.
+From this [[Proof by Contradiction|contradiction]] follows the result.
+{{qed}}
+{{Euclid Note|7|VIII}}
+\end{proof}<|endoftext|>
+\section{Geometric Sequences in Proportion have Same Number of Elements}
+Tags: Geometric Sequences of Integers
+
+\begin{theorem}
+Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]] of [[Definition:Length of Sequence|length]] $n$.
+Let $r$ be the [[Definition:Common Ratio|common ratio]] of $P$.
+Let $Q = \sequence {b_j}_{0 \mathop \le j \mathop \le m}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]] of [[Definition:Length of Sequence|length]] $m$.
+Let $r$ be the [[Definition:Common Ratio|common ratio]] of $Q$.
+Let $b_0$ and $b_m$ be such that $\dfrac {b_0} {b_m} = \dfrac {a_0} {a_n}$.
+Then $m = n$.
+{{:Euclid:Proposition/VIII/8}}
+\end{theorem}
+
+\begin{proof}
+Let $S = \sequence {c_j}_{0 \mathop \le j \mathop \le n}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]] of [[Definition:Length of Sequence|length]] $n$ such that:
+:$r$ is the [[Definition:Common Ratio|common ratio]] of $S$
+:$S$ is in its [[Definition:Geometric Sequence of Integers in Lowest Terms|lowest terms]].
+From [[Geometric Sequence in Lowest Terms has Coprime Extremes]], $c_0$ is [[Definition:Coprime Integers|coprime]] to $c_n$.
+Then:
+:$\dfrac {c_0} {c_n} = \dfrac {a_0} {a_n} = \dfrac {b_0} {b_m}$
+But:
+:$c_n = r^n c_0$
+and so:
+:$b_m = r^n b_0$
+The result follows.
+{{qed}}
+{{Euclid Note|8|VIII}}
+\end{proof}<|endoftext|>
+\section{Elements of Geometric Sequence between Coprime Numbers}
+Tags: Geometric Sequences of Integers
+
+\begin{theorem}
+Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]] of [[Definition:Length of Sequence|length]] $n + 1$.
+Let $a_0$ be [[Definition:Coprime Integers|coprime]] to $a_n$.
+Then there exist [[Definition:Geometric Sequence of Integers|geometric sequences of integers]] $Q_1$ and $Q_2$ of [[Definition:Length of Sequence|length]] $n + 1$ such that:
+:the [[Definition:Initial Term of Geometric Sequence|initial term]] of both $Q_1$ and $Q_2$ is $1$
+:the [[Definition:Final Term of Geometric Sequence|final term]] of $Q_1$ is $a_0$
+:the [[Definition:Final Term of Geometric Sequence|final term]] of $Q_2$ is $a_n$.
+{{:Euclid:Proposition/VIII/9}}
+\end{theorem}
+
+\begin{proof}
+Let the [[Definition:Common Ratio|common ratio]] of $P$ be $r$.
+By [[Form of Geometric Sequence of Integers with Coprime Extremes]], the $j$th [[Definition:Term of Geometric Sequence|term]] of $P$ is given by:
+:$a_j = q^j p^{n - j}$
+such that:
+:$a_0 = p^n$
+:$a_n = q^n$
+For $j \in \set {0, 1, \ldots, n}$, let $r_j = q^j$.
+Let the [[Definition:Finite Sequence|finite sequence]] $Q_2 = \sequence {t_j}_{0 \mathop \le j \mathop \le n}$ be defined as:
+:$\forall j \in \set {0, 1, \ldots, n}: t_j = p^{n - j}$
+Then $Q_1$ is a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]] of [[Definition:Length of Sequence|length]] $n$ such that:
+:$t_0 = 1$
+:$t_n = p^n$
+Let the [[Definition:Finite Sequence|finite sequence]] $Q_1 = \sequence {s_j}_{0 \mathop \le j \mathop \le n}$ be defined as:
+:$\forall j \in \set {0, 1, \ldots, n}: s_j = q^j$
+Then $Q_1$ is a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]] of [[Definition:Length of Sequence|length]] $n$ such that:
+:$s_0 = 1$
+:$s_n = q^n$
+Hence the result.
+{{qed}}
+{{Euclid Note|9|VIII|{{EuclidNoteConverse|title = Product of Geometric Sequences from One|book = VIII|prop = 10}}}}
+\end{proof}<|endoftext|>
+\section{Product of Geometric Sequences from One}
+Tags: Geometric Sequences of Integers
+
+\begin{theorem}
+Let $Q_1 = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ and $Q_2 = \sequence {b_j}_{0 \mathop \le j \mathop \le n}$ be [[Definition:Geometric Sequence of Integers|geometric sequences of integers]] of [[Definition:Length of Sequence|length]] $n + 1$.
+Let $a_0 = b_0 = 1$.
+Then the [[Definition:Finite Sequence|sequence]] $P = \sequence {p_j}_{0 \mathop \le j \mathop \le n}$ defined as:
+:$\forall j \in \set {0, \ldots, n}: p_j = b^j a^{n - j}$
+is a [[Definition:Geometric Sequence|geometric sequence]].
+{{:Euclid:Proposition/VIII/10}}
+\end{theorem}
+
+\begin{proof}
+By [[Form of Geometric Sequence of Integers with Coprime Extremes]], the $j$th [[Definition:Term of Geometric Sequence|term]] of $Q_1$ is given by:
+:$a_j = a^j$
+and of $Q_1$ is given by:
+:$b_j = b^j$
+Let the [[Definition:Term of Geometric Sequence|terms]] of $P$ be defined as:
+:$\forall j \in \left\{{0, 1, \ldots, n}\right\}: p_j = b^j a^{n - j}$
+Then from [[Form of Geometric Sequence of Integers]] it follows that $P$ is a [[Definition:Geometric Sequence|geometric sequence]].
+{{qed}}
+{{Euclid Note|10|VIII|{{EuclidNoteConverse|title = Elements of Geometric Sequence between Coprime Numbers|book = VIII|prop = 9}}}}
+\end{proof}<|endoftext|>
+\section{Powers of Elements of Geometric Sequence are in Geometric Sequence}
+Tags: Geometric Sequences of Integers
+
+\begin{theorem}
+Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]].
+Then the [[Definition:Finite Sequence|sequence]] $Q = \sequence {b_j}_{0 \mathop \le j \mathop \le n}$ defined as:
+:$\forall j \in \set {0, 1, \ldots, n}: b_j = a_j^m$
+where $m \in \Z_{>0}$, is a [[Definition:Geometric Sequence of Integers|geometric sequence]].
+{{:Euclid:Proposition/VIII/13}}
+\end{theorem}
+
+\begin{proof}
+From [[Form of Geometric Sequence of Integers]], the $j$th [[Definition:Term of Geometric Sequence|term]] of $P$ is given by:
+:$a_j = k q^j p^{n - j}$
+Thus the $j$th [[Definition:Term of Geometric Sequence|term]] of $Q$ is given by:
+:$b_j = k^m \paren {q^m}^j \paren {p^m}^{n - j}$
+From [[Form of Geometric Sequence of Integers]], this is a [[Definition:Geometric Sequence of Integers|geometric sequence]].
+{{qed}}
+{{Euclid Note|13|VIII}}
+\end{proof}<|endoftext|>
+\section{Number divides Number iff Square divides Square}
+Tags: Divisors, Square Numbers
+
+\begin{theorem}
+Let $a, b \in \Z$.
+Then:
+:$a^2 \divides b^2 \iff a \divides b$
+where $\divides$ denotes [[Definition:Divisor of Integer|integer divisibility]].
+{{:Euclid:Proposition/VIII/14}}
+\end{theorem}
+
+\begin{proof}
+From [[Between two Squares exists one Mean Proportional]]:
+:$\tuple {a^2, ab, b^2}$
+is a [[Definition:Geometric Sequence of Integers|geometric sequence]].
+Let $a, b \in \Z$ such that $a^2 \divides b^2$.
+Then from [[First Element of Geometric Sequence that divides Last also divides Second]]:
+:$a^2 \divides a b$
+Thus:
+{{begin-eqn}}
+{{eqn | l = a^2
+      | o = \divides
+      | r = a b
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | lo= \exists k \in \Z:
+      | l = k a^2
+      | r = a b
+      | c = {{Defof|Divisor of Integer}}
+}}
+{{eqn | ll= \leadsto
+      | l = k a
+      | r = b
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = a
+      | o = \divides
+      | r = b
+      | c = {{Defof|Divisor of Integer}}
+}}
+{{end-eqn}}
+{{qed|lemma}}
+Let $a \divides b$.
+Then:
+{{begin-eqn}}
+{{eqn | l = a
+      | o = \divides
+      | r = b
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | lo= \exists k \in \Z:
+      | l = k a
+      | r = b
+      | c = {{Defof|Divisor of Integer}}
+}}
+{{eqn | ll= \leadsto
+      | l = k a^2
+      | r = a b
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = k a b
+      | r = b^2
+      | c = {{Defof|Geometric Sequence of Integers}}
+}}
+{{eqn | ll= \leadsto
+      | l = k^2 a^2
+      | r = b^2
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = a^2
+      | o = \divides
+      | r = b^2
+      | c = {{Defof|Divisor of Integer}}
+}}
+{{end-eqn}}
+{{qed}}
+{{Euclid Note|14|VIII}}
+\end{proof}<|endoftext|>
+\section{Number divides Number iff Cube divides Cube}
+Tags: Divisors, Cube Numbers
+
+\begin{theorem}
+Let $a, b \in \Z$.
+Then:
+:$a^3 \divides b^3 \iff a \divides b$
+where $\divides$ denotes [[Definition:Divisor of Integer|integer divisibility]].
+{{:Euclid:Proposition/VIII/15}}
+\end{theorem}
+
+\begin{proof}
+Let $a^3$ and $b^3$ be [[Definition:Cube Number|cube numbers]].
+From [[Form of Geometric Sequence of Integers/Corollary|the corollary to Form of Geometric Sequence of Integers]]:
+:$\tuple {a^3, a^2 b, a b^2, b^3}$
+is a [[Definition:Geometric Sequence of Integers|geometric sequence]].
+Let $a, b \in \Z$ such that $a^2 \divides b^2$.
+Then from [[First Element of Geometric Sequence that divides Last also divides Second]]:
+:$a^3 \divides a^2 b$
+Thus:
+{{begin-eqn}}
+{{eqn | l = a^3
+      | o = \divides
+      | r = a^2 b
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | lo= \exists k \in \Z:
+      | l = k a^3
+      | r = a^2 b
+      | c = {{Defof|Divisor of Integer}}
+}}
+{{eqn | ll= \leadsto
+      | l = k a
+      | r = b
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = a
+      | o = \divides
+      | r = b
+      | c = {{Defof|Divisor of Integer}}
+}}
+{{end-eqn}}
+{{qed|lemma}}
+Let $a \divides b$.
+Then:
+{{begin-eqn}}
+{{eqn | l = a
+      | o = \divides
+      | r = b
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | lo= \exists k \in \Z:
+      | l = k a
+      | r = b
+      | c = {{Defof|Divisor of Integer}}
+}}
+{{eqn | ll= \leadsto
+      | l = k a^3
+      | r = a^2 b
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = a^3
+      | o = \divides
+      | r = a^2 b
+      | c = {{Defof|Divisor of Integer}}
+}}
+{{eqn | ll= \leadsto
+      | l = a^3
+      | o = \divides
+      | r = b^3
+      | c = [[Divisibility of Elements in Geometric Sequence of Integers]]
+}}
+{{end-eqn}}
+{{qed}}
+{{Euclid Note|15|VIII}}
+\end{proof}<|endoftext|>
+\section{Number does not divide Number iff Square does not divide Square}
+Tags: Divisors, Square Numbers
+
+\begin{theorem}
+Let $a, b \in \Z$ be [[Definition:Integer|integers]].
+Then:
+:$a \nmid b \iff a^2 \nmid b^2$
+where $a \nmid b$ denotes that $a$ is not a [[Definition:Divisor of Integer|divisor]] of $b$.
+{{:Euclid:Proposition/VIII/16}}
+\end{theorem}
+
+\begin{proof}
+Let $a \nmid b$.
+{{AimForCont}}:
+:$a^2 \divides b^2$
+where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
+Then by [[Number divides Number iff Square divides Square]]:
+:$a \divides b$
+From [[Proof by Contradiction]] it follows that $a^2 \divides b^2$ is [[Definition:False|false]].
+Thus $a^2 \nmid b^2$.
+{{qed|lemma}}
+Let $a^2 \nmid b^2$.
+{{AimForCont}}:
+:$a \divides b$
+Then by [[Number divides Number iff Square divides Square]]:
+:$a^2 \divides b^2$
+From [[Proof by Contradiction]] it follows that $a \divides b$ is [[Definition:False|false]].
+Thus $a \nmid b$.
+{{qed}}
+{{Euclid Note|16|VIII}}
+\end{proof}<|endoftext|>
+\section{Number does not divide Number iff Cube does not divide Cube}
+Tags: Divisors, Cube Numbers
+
+\begin{theorem}
+Let $a, b \in \Z$ be [[Definition:Integer|integers]].
+Then:
+:$a \nmid b \iff a^3 \nmid b^3$
+where $a \nmid b$ denotes that $a$ is not a [[Definition:Divisor of Integer|divisor]] of $b$.
+{{:Euclid:Proposition/VIII/17}}
+\end{theorem}
+
+\begin{proof}
+Let $a \nmid b$.
+{{AimForCont}}:
+:$a^3 \divides b^3$
+where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
+Then by [[Number divides Number iff Cube divides Cube]]:
+:$a \divides b$
+From [[Proof by Contradiction]] it follows that $a^2 \divides b^2$ is [[Definition:False|false]].
+Thus $a^3 \nmid b^3$.
+{{qed|lemma}}
+Let $a^3 \nmid b^3$.
+{{AimForCont}}
+:$a \divides b$
+Then by [[Number divides Number iff Cube divides Cube]]:
+:$a^3 \divides b^3$
+From [[Proof by Contradiction]] it follows that $a \divides b$ is [[Definition:False|false]].
+Thus $a \nmid b$.
+{{qed}}
+{{Euclid Note|17|VIII}}
+\end{proof}<|endoftext|>
+\section{Numbers between which exists one Mean Proportional are Similar Plane}
+Tags: Euclidean Number Theory
+
+\begin{theorem}
+Let $a, b \in \Z$ such that the [[Definition:Geometric Mean|geometric mean]] is an [[Definition:Integer|integer]].
+Then $a$ and $b$ are [[Definition:Similar Plane Numbers|similar plane numbers]].
+{{:Euclid:Proposition/VIII/20}}
+\end{theorem}
+
+\begin{proof}
+Let the [[Definition:Geometric Mean|geometric mean]] of $a$ and $b$ be an [[Definition:Integer|integer]] $m$.
+Then, in the language of {{AuthorRef|Euclid}}, $m$ is a [[Definition:Mean Proportional|mean proportional]] of $a$ and $b$.
+Thus:
+:$\left({a, m, b}\right)$
+is a [[Definition:Geometric Sequence of Integers|geometric sequence]].
+From [[Form of Geometric Sequence of Integers]]:
+:$\exists k, p, q \in \Z: a = k p^2, b = k q^2$
+So $a$ and $b$ are [[Definition:Plane Number|plane numbers]] whose sides are:
+:$k p$ and $p$
+and
+:$k q$ and $q$
+respectively.
+Then:
+:$\dfrac {k p} {k q} = \dfrac p q$
+demonstrating that $a$ and $b$ are [[Definition:Similar Plane Numbers|similar plane numbers]] by definition.
+{{qed}}
+{{Euclid Note|20|VIII}}
+\end{proof}<|endoftext|>
+\section{Numbers between which exist two Mean Proportionals are Similar Solid}
+Tags: Euclidean Number Theory
+
+\begin{theorem}
+Let $a, b \in \Z$ be the [[Definition:Extremes of Finite Geometric Sequence|extremes]] of a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]] whose [[Definition:Length of Sequence|length]] is $4$:
+:$\tuple {a, m_1, m_2, b}$
+That is, such that $a$ and $b$ have $2$ [[Definition:General Mean Proportional|mean proportionals]].
+Then $a$ and $b$ are [[Definition:Similar Solid Numbers|similar solid numbers]].
+{{:Euclid:Proposition/VIII/21}}
+\end{theorem}
+
+\begin{proof}
+From [[Form of Geometric Sequence of Integers]]:
+:$\exists k, p, q \in \Z: a = k p^3, b = k q^3$
+So $a$ and $b$ are [[Definition:Solid Number|solid numbers]] whose sides are:
+:$k p$, $p$ and $p$
+and
+:$k q$, $q$ and $q$
+respectively.
+Then:
+:$\dfrac {k p} {k q} = \dfrac p q$
+demonstrating that $a$ and $b$ are [[Definition:Similar Solid Numbers|similar solid numbers]] by definition.
+{{qed}}
+{{Euclid Note|21|VIII}}
+\end{proof}<|endoftext|>
+\section{If First of Three Numbers in Geometric Sequence is Square then Third is Square}
+Tags: Geometric Sequences of Integers
+
+\begin{theorem}
+Let $P = \tuple {a, b, c}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]].
+Let $a$ be a [[Definition:Square Number|square number]].
+Then $c$ is also a [[Definition:Square Number|square number]].
+{{:Euclid:Proposition/VIII/22}}
+\end{theorem}
+
+\begin{proof}
+From [[Form of Geometric Sequence of Integers]]:
+:$P = \tuple {k p^2, k p q, k q^2}$
+for some $k, p, q \in \Z$.
+If $a = k p^2$ is a [[Definition:Square Number|square number]] it follows that $k$ is a [[Definition:Square Number|square number]]: $k = r^2$, say.
+So:
+:$P = \tuple {r^2 p^2, r^2 p q, r^2 q^2}$
+and so $c = r^2 q^2 = \paren {r q}^2$.
+{{qed}}
+{{Euclid Note|22|VIII}}
+\end{proof}<|endoftext|>
+\section{If First of Four Numbers in Geometric Sequence is Cube then Fourth is Cube}
+Tags: Geometric Sequences of Integers
+
+\begin{theorem}
+Let $P = \tuple {a, b, c, d}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]].
+Let $a$ be a [[Definition:Cube Number|cube number]].
+Then $d$ is also a [[Definition:Cube Number|cube number]].
+{{:Euclid:Proposition/VIII/23}}
+\end{theorem}
+
+\begin{proof}
+From [[Form of Geometric Sequence of Integers]]:
+:$P = \tuple {k p^3, k p^2 q, k p q^2, k q^3}$
+for some $k, p, q \in \Z$.
+If $a = k p^3$ is a [[Definition:Cube Number|cube number]] it follows that $k$ is a [[Definition:Cube Number|cube number]]: $k = r^3$, say.
+So:
+:$P = \tuple {r^3 p^3, r^3 p^2 q, r^3 p q^2, r^3 q^3}$
+and so $d = r^3 q^3 = \paren {r q}^3$.
+{{qed}}
+{{Euclid Note|23|VIII}}
+\end{proof}<|endoftext|>
+\section{If Ratio of Square to Number is as between Two Squares then Number is Square}
+Tags: Ratios, Square Numbers
+
+\begin{theorem}
+Let $a, b, c, d \in \Z$ be [[Definition:Integer|integers]] such that:
+:$\dfrac a b = \dfrac {c^2} {d^2}$
+Let $a$ be a [[Definition:Square Number|square number]].
+Then $b$ is also a [[Definition:Square Number|square number]].
+{{:Euclid:Proposition/VIII/24}}
+\end{theorem}
+
+\begin{proof}
+From {{EuclidPropLink|title = Between two Similar Plane Numbers exists one Mean Proportional|book = VIII|prop = 18}}:
+:$\tuple {c^2, c d, d^2}$
+is a [[Definition:Geometric Sequence of Integers|geometric sequence]].
+From {{EuclidPropLink|title = Geometric Sequences in Proportion have Same Number of Elements|book = VIII|prop = 8}}:
+:$\tuple {a, m, b}$
+is a [[Definition:Geometric Sequence of Integers|geometric sequence]] for some $m$.
+We have that $a$ is a [[Definition:Square Number|square number]].
+From {{EuclidPropLink|title = If First of Three Numbers in Geometric Sequence is Square then Third is Square|book = VIII|prop = 22}}:
+:$b$ is a [[Definition:Square Number|square number]].
+{{qed}}
+{{Euclid Note|24|VIII}}
+\end{proof}<|endoftext|>
+\section{If Ratio of Cube to Number is as between Two Cubes then Number is Cube}
+Tags: Ratios, Cube Numbers
+
+\begin{theorem}
+Let $a, b, c, d \in \Z$ be [[Definition:Integer|integers]] such that:
+:$\dfrac a b = \dfrac {c^3} {d^3}$
+Let $a$ be a [[Definition:Cube Number|cube number]].
+Then $b$ is also a [[Definition:Cube Number|cube number]].
+{{:Euclid:Proposition/VIII/25}}
+\end{theorem}
+
+\begin{proof}
+From {{EuclidPropLink|title = Between two Similar Solid Numbers exist two Mean Proportionals|book = VIII|prop = 19}}:
+:$\left({c^3, c^2 d, c d^2, d^3}\right)$
+is a [[Definition:Geometric Sequence of Integers|geometric sequence]].
+From {{EuclidPropLink|title = Geometric Sequences in Proportion have Same Number of Elements|book = VIII|prop = 8}}:
+:$\left({a, m_1, m_2, b}\right)$
+is a [[Definition:Geometric Sequence of Integers|geometric sequence]] for some $m$.
+We have that $a$ is a [[Definition:Cube Number|cube number]].
+From {{EuclidPropLink|title = If First of Four Numbers in Geometric Sequence is Cube then Fourth is Cube|book = VIII|prop = 23}}:
+:$b$ is a [[Definition:Cube Number|cube number]].
+{{qed}}
+{{Euclid Note|25|VIII}}
+\end{proof}<|endoftext|>
+\section{Square of Cube Number is Cube}
+Tags: Euclidean Number Theory, Square of Cube Number is Cube
+
+\begin{theorem}
+Let $a \in \N$ be a [[Definition:Natural Number|natural number]].
+Let $a$ be a [[Definition:Cube Number|cube number]].
+Then $a^2$ is also a [[Definition:Cube Number|cube number]].
+{{:Euclid:Proposition/IX/3}}
+\end{theorem}
+
+\begin{proof}
+By the definition of [[Definition:Cube Number|cube number]]:
+:$\exists k \in \N: k^3 = a$
+Thus:
+{{begin-eqn}}
+{{eqn | l = a^2
+      | r = \paren {k^3}^2
+      | c = 
+}}
+{{eqn | r = k^6
+      | c = 
+}}
+{{eqn | r = \paren {k^2}^3
+      | c = 
+}}
+{{end-eqn}}
+Thus:
+:$\exists r = k^2 \in \N: a = r^3$
+Hence the result by definition of [[Definition:Cube Number|cube number]].
+{{qed}}
+\end{proof}
+
+\begin{proof}
+From [[Cube Number multiplied by Cube Number is Cube]], if $a$ and $b$ are [[Definition:Cube Number|cube numbers]] then $a b$ is a [[Definition:Cube Number|cube number]].
+The result follows by setting $b = a$.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Cube Number multiplied by Cube Number is Cube}
+Tags: Euclidean Number Theory
+
+\begin{theorem}
+Let $a, b \in \N$ be [[Definition:Natural Number|natural numbers]].
+Let $a$ and $b$ be [[Definition:Cube Number|cube numbers]].
+Then $a b$ is also a [[Definition:Cube Number|cube number]].
+{{:Euclid:Proposition/IX/4}}
+\end{theorem}
+
+\begin{proof}
+By the definition of [[Definition:Cube Number|cube number]]:
+:$\exists r \in \N: r^3 = a$
+:$\exists s \in \N: s^3 = b$
+Thus:
+{{begin-eqn}}
+{{eqn | l = a b
+      | r = \paren {r^3} \paren {s^3}
+      | c = [[Exponent Combination Laws/Power of Product|Exponent Combination Laws: Power of Product]]
+}}
+{{eqn | r = \paren {r s}^3
+      | c = 
+}}
+{{end-eqn}}
+Thus:
+:$\exists k = r s \in \N: a = k^3$
+Hence the result by definition of [[Definition:Cube Number|cube number]].
+{{qed}}
+{{Euclid Note|4|IX|The proof as given here is not that given by {{AuthorRef|Euclid}}, as the latter is unwieldy and of limited use.}}
+\end{proof}<|endoftext|>
+\section{Number Squared making Cube is itself Cube}
+Tags: Euclidean Number Theory
+
+\begin{theorem}
+Let $a \in \Z$ be an [[Definition:Integer|integer]].
+Let $a^2$ be a [[Definition:Cube Number|cube number]].
+Then $a$ is a [[Definition:Cube Number|cube number]].
+{{:Euclid:Proposition/IX/6}}
+\end{theorem}
+
+\begin{proof}
+$a^2$ and $a^3$ are both [[Definition:Cube Number|cube numbers]].
+From [[Between two Cubes exist two Mean Proportionals]] the [[Definition:Finite Sequence|sequence]]:
+:$a^3, m_1, m_2, a^2$
+is a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]] for some $m_1, m_2 \in \Z$.
+By [[Geometric Sequences in Proportion have Same Number of Elements]]:
+:$a^2, m_3, m_4, a$
+is a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]] for some $m_3, m_4 \in \Z$.
+From [[If First of Four Numbers in Geometric Sequence is Cube then Fourth is Cube]] it follows that $a$ is a [[Definition:Cube Number|cube number]].
+{{qed}}
+{{Euclid Note|6|IX}}
+\end{proof}<|endoftext|>
+\section{Product of Composite Number with Number is Solid Number}
+Tags: Euclidean Number Theory
+
+\begin{theorem}
+Let $a, b \in \Z$ be [[Definition:Positive Integer|positive integers]].
+Let $a$ be a [[Definition:Composite Number|composite number]].
+Then $a b$ is a [[Definition:Solid Number|solid number]].
+{{:Euclid:Proposition/IX/7}}
+\end{theorem}
+
+\begin{proof}
+By definition of [[Definition:Composite Number|composite number]]:
+:$\exists p, q \in \Z_{>1}: a = p q$
+Then:
+:$a b = p q b$
+Hence the result by definition of [[Definition:Solid Number|solid number]].
+{{qed}}
+{{Euclid Note|7|IX}}
+\end{proof}<|endoftext|>
+\section{Square Numbers are Similar Plane Numbers}
+Tags: Euclidean Number Theory
+
+\begin{theorem}
+Let $a$ and $b$ be [[Definition:Square Number|square numbers]].
+Then $a$ and $b$ are [[Definition:Similar Plane Numbers|similar plane numbers]].
+\end{theorem}
+
+\begin{proof}
+Let $a = c^2$ and $b = d^2$.
+We have that:
+: $c : d = c : d$
+The result follows by definition of [[Definition:Similar Plane Numbers|similar plane numbers]]
+{{qed}}
+[[Category:Euclidean Number Theory]]
+r9mhpqv1z04e43xu2c1t037yxl8odoh
+\end{proof}<|endoftext|>
+\section{Elements of Geometric Sequence from One which are Powers of Number}
+Tags: Geometric Sequences of Integers
+
+\begin{theorem}
+Let $G_n = \sequence {a_n}_{0 \mathop \le i \mathop \le n}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]].
+Let $a_0 = 1$.
+Then:
+:$\forall m, k \in \set {1, \ldots, n}: k \divides m \implies a_m$ is a [[Definition:Integer Power|power]] of $k$
+where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
+{{:Euclid:Proposition/IX/8}}
+\end{theorem}
+
+\begin{proof}
+By [[Form of Geometric Sequence of Integers from One]], the general [[Definition:Term of Geometric Sequence|term]] of $G_n$ can be expressed as:
+:$a_j = q^j$
+for some $q \in \Z$.
+Let $k, m \in \set {1, 2, \ldots, n}$ such that $k \divides m$.
+By definition of [[Definition:Divisor of Integer|divisibility]]:
+:$\exists r \in \Z: m = r k$
+Then:
+{{begin-eqn}}
+{{eqn | l = a_m
+      | r = q^m
+      | c = 
+}}
+{{eqn | r = q^{r k}
+      | c = 
+}}
+{{eqn | r = \paren {q^r}^k
+      | c = 
+}}
+{{end-eqn}}
+That is, $a_m$ is a [[Definition:Integer Power|power]] of $k$.
+{{qed}}
+{{Euclid Note|8|IX}}
+\end{proof}<|endoftext|>
+\section{Elements of Geometric Sequence from One where First Element is Power of Number}
+Tags: Geometric Sequences
+
+\begin{theorem}
+Let $G_n = \sequence {a_n}_{0 \mathop \le i \mathop \le n}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]].
+Let $a_0 = 1$.
+Let $m \in \Z_{> 0}$.
+Let $a_1$ be the $m$th [[Definition:Integer Power|power]] of an [[Definition:Integer|integer]].
+Then all the [[Definition:Term of Geometric Sequence|terms]] of $G_n$ are $m$th [[Definition:Integer Power|powers]] of  [[Definition:Integer|integers]].
+{{:Euclid:Proposition/IX/9}}
+\end{theorem}
+
+\begin{proof}
+By [[Form of Geometric Sequence of Integers from One]], the general [[Definition:Term of Geometric Sequence|term]] of $G_n$ can be expressed as:
+:$a_j = q^j$
+for some $q \in \Z$.
+Let $a_2 = k^m$.
+By definition of [[Definition:Geometric Sequence|geometric sequence]]:
+:$\forall j \in \set {1, 2, \ldots, n}: a_j = r a_{j - 1}$
+where $r$ is the [[Definition:Common Ratio|common ratio]].
+This holds specifically for $j = 1$:
+:$k^m = r \cdot 1$
+Thus:
+{{begin-eqn}}
+{{eqn | l = a_j
+      | r = \paren {k^m}^j
+      | c = 
+}}
+{{eqn | r = k^{m j}
+      | c = 
+}}
+{{eqn | r = \paren {k^j}^m
+      | c = 
+}}
+{{end-eqn}}
+Hence the result.
+{{qed}}
+{{Euclid Note|9|IX}}
+\end{proof}<|endoftext|>
+\section{Elements of Geometric Sequence from One where First Element is not Power of Number}
+Tags: Geometric Sequences
+
+\begin{theorem}
+Let $G_n = \sequence {a_n}_{0 \mathop \le i \mathop \le n}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]].
+Let $a_0 = 1$.
+Let $k \in \Z_{> 1}$.
+Let $a_1$ not be a [[Definition:Integer Power|power]] of $k$.
+Then $a_m$ is not a [[Definition:Integer Power|power]] of $k$ except for:
+:$\forall m, k \in \set {1, 2, \ldots, n}: k \divides m$
+where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
+{{:Euclid:Proposition/IX/10}}
+\end{theorem}
+
+\begin{proof}
+By [[Form of Geometric Sequence of Integers from One]], the general [[Definition:Term of Geometric Sequence|term]] of $G_n$ can be expressed as:
+:$a_j = q^j$
+for some $q \in \Z$.
+Let $k \nmid m$.
+Then by the [[Division Theorem]] there exists a unique $q \in \Z$ such that:
+:$m = k q + b$
+for some $b$ such that $0 < b < k$.
+Thus:
+:$a_m = a^{k q} a^b$
+which is not a [[Definition:Integer Power|power]] of $k$.
+{{qed}}
+{{Euclid Note|10|IX}}
+\end{proof}<|endoftext|>
+\section{Elements of Geometric Sequence from One which Divide Later Elements}
+Tags: Geometric Sequences
+
+\begin{theorem}
+Let $G_n = \sequence {a_n}_{0 \mathop \le i \mathop \le n}$ be a [[Definition:Geometric Sequence of Integers|geometric Sequence of integers]].
+Let $a_0 = 1$.
+Let $m \in \Z_{> 0}$.
+Then:
+:$\forall r \in \set {0, 1, \ldots, m}: a_k \divides a_m$
+where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
+{{:Euclid:Proposition/IX/11}}
+\end{theorem}
+
+\begin{proof}
+By [[Form of Geometric Sequence of Integers from One]], the general [[Definition:Term of Geometric Sequence|term]] of $G_n$ can be expressed as:
+:$a_j = q^j$
+for some $q \in \Z$.
+Hence the result from [[Divisors of Power of Prime]].
+{{qed}}
+{{Euclid Note|11|IX}}
+\end{proof}<|endoftext|>
+\section{Elements of Geometric Sequence from One Divisible by Prime}
+Tags: Geometric Sequences
+
+\begin{theorem}
+Let $G_n = \sequence {a_n}_{0 \mathop \le i \mathop \le n}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]].
+Let $a_0 = 1$.
+Let $p$ be a [[Definition:Prime Number|prime number]] such that:
+:$p \divides a_n$
+where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
+Then $p \divides a_1$.
+{{:Euclid:Proposition/IX/12}}
+\end{theorem}
+
+\begin{proof}
+By [[Form of Geometric Sequence of Integers from One]], the general [[Definition:Term of Geometric Sequence|term]] of $G_n$ can be expressed as:
+:$a_j = q^j$
+for some $q \in \Z$.
+Thus [[Definition:By Hypothesis|by hypothesis]]:
+:$p \divides q^n$
+From [[Euclid's Lemma for Prime Divisors/General Result|Euclid's Lemma for Prime Divisors: General Result]]:
+:$p \divides q$
+Hence the result.
+{{qed}}
+{{Euclid Note|12|IX}}
+\end{proof}<|endoftext|>
+\section{Divisibility of Elements of Geometric Sequence from One where First Element is Prime}
+Tags: Geometric Sequences of Integers
+
+\begin{theorem}
+Let $Q_n = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a [[Definition:Geometric Sequence|geometric sequence]] of [[Definition:Length of Sequence|length]] $n$ consisting of [[Definition:Integer|integers]] only.
+Let $a_0 = 1$.
+Let $a_1$ be a [[Definition:Prime Number|prime number]].
+Then the only [[Definition:Divisor of Integer|divisors]] of $a_n$ are $a_j$ for $j \in \set {1, 2, \ldots, n}$.
+{{:Euclid:Proposition/IX/13}}
+\end{theorem}
+
+\begin{proof}
+From [[Form of Geometric Sequence of Integers from One]], the elements of $Q_n$ are given by:
+:$Q_n = \tuple {1, a, a^2, \ldots, a^n}$
+From [[Divisors of Power of Prime]], each of $a_j$ for $j \in \set {1, 2, \ldots, n}$ are the only [[Definition:Divisor of Integer|divisors]] of $a_n$.
+{{qed}}
+{{Euclid Note|13|IX}}
+\end{proof}<|endoftext|>
+\section{Sum of Pair of Elements of Geometric Sequence with Three Elements in Lowest Terms is Coprime to other Element}
+Tags: Geometric Sequences
+
+\begin{theorem}
+Let $P = \tuple {a, b, c}$ be a [[Definition:Geometric Sequence of Integers in Lowest Terms|geometric sequence of integers in its lowest terms]].
+Then $\paren {a + b}$, $\paren {b + c}$ and $\paren {a + c}$ are all [[Definition:Coprime Integers|coprime]] to each of $a$, $b$ and $c$.
+{{:Euclid:Proposition/IX/15}}
+\end{theorem}
+
+\begin{proof}
+Let the [[Definition:Common Ratio|common ratio]] of $P$ in [[Definition:Canonical Form of Rational Number|canonical form]] be $\dfrac q p$.
+By [[Form of Geometric Sequence of Integers in Lowest Terms]]:
+:$P = \tuple {p^2, p q, q^2}$
+Then:
+{{begin-eqn}}
+{{eqn | l = p
+      | o = \perp
+      | r = q
+      | c = {{Defof|Canonical Form of Rational Number}}
+}}
+{{eqn | ll= \leadsto
+      | l = q
+      | o = \perp
+      | r = \paren {p + q}
+      | c = [[Numbers are Coprime iff Sum is Coprime to Both]]
+}}
+{{eqn | ll= \leadsto
+      | l = q
+      | o = \perp
+      | r = p \paren {p + q}
+      | c = [[Integer Coprime to all Factors is Coprime to Whole]]
+}}
+{{eqn | ll= \leadsto
+      | l = q^2
+      | o = \perp
+      | r = p \paren {p + q}
+      | c = [[Square of Coprime Number is Coprime]]
+}}
+{{eqn | ll= \leadsto
+      | l = q^2
+      | o = \perp
+      | r = p^2 + p q
+      | c = factorising
+}}
+{{eqn | ll= \leadsto
+      | l = c
+      | o = \perp
+      | r = a + b
+      | c = 
+}}
+{{end-eqn}}
+Similarly:
+{{begin-eqn}}
+{{eqn | ll= \leadsto
+      | l = p^2
+      | o = \perp
+      | r = p q + q^2
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = a
+      | o = \perp
+      | r = b + c
+      | c = factorising
+}}
+{{end-eqn}}
+Then:
+{{begin-eqn}}
+{{eqn | l = p + q
+      | o = \perp
+      | r = p
+      | c = [[Numbers are Coprime iff Sum is Coprime to Both]]
+}}
+{{eqn | lo= \land
+      | l = p + q
+      | o = \perp
+      | r = q
+      | c = [[Numbers are Coprime iff Sum is Coprime to Both]]
+}}
+{{eqn | ll= \leadsto
+      | l = \paren {p + q}^2
+      | o = \perp
+      | r = p q
+      | c = as above
+}}
+{{eqn | ll= \leadsto
+      | l = p^2 + q^2 + 2 p q
+      | o = \perp
+      | r = p q
+      | c = factorising
+}}
+{{eqn | ll= \leadsto
+      | l = p^2 + q^2
+      | o = \perp
+      | r = p q
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = b
+      | o = \perp
+      | r = a + c
+      | c = 
+}}
+{{end-eqn}}
+{{qed}}
+{{Euclid Note|15|IX}}
+\end{proof}<|endoftext|>
+\section{Two Coprime Integers have no Third Integer Proportional}
+Tags: Coprime Integers, Ratios
+
+\begin{theorem}
+Let $a, b \in \Z_{>0}$ be [[Definition:Integer|integers]] such that $a$ and $b$ are [[Definition:Coprime Integers|coprime]].
+Then there is no [[Definition:Integer|integer]] $c \in \Z$ such that:
+:$\dfrac a b = \dfrac b c$
+{{:Euclid:Proposition/IX/16}}
+\end{theorem}
+
+\begin{proof}
+Suppose such a $c$ exists.
+From [[Coprime Numbers form Fraction in Lowest Terms]], $\dfrac a b$ is in [[Definition:Canonical Form of Rational Number|canonical form]].
+From [[Ratios of Fractions in Lowest Terms]]:
+:$a \divides b$
+where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
+This contradicts the fact that $a$ and $b$ are [[Definition:Coprime Integers|coprime]].
+Hence such a $c$ cannot exist.
+{{qed}}
+{{Euclid Note|16|IX}}
+\end{proof}<|endoftext|>
+\section{Last Element of Geometric Sequence with Coprime Extremes has no Integer Proportional as First to Second}
+Tags: Geometric Sequences
+
+\begin{theorem}
+Let $G_n = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence of integers]] such that $a_0 \ne 1$.
+Let $a_0 \perp a_n$, where $\perp$ denotes [[Definition:Coprime Integers|coprimality]].
+Then there does not exist an [[Definition:Integer|integer]] $b$ such that:
+:$\dfrac {a_0} {a_1} = \dfrac {a_n} b$
+{{:Euclid:Proposition/IX/17}}
+\end{theorem}
+
+\begin{proof}
+{{AimForCont}} there exists $b$ such that $\dfrac {a_0} {a_1} = \dfrac {a_n} b$.
+Then:
+:$\dfrac {a_0} {a_n} = \dfrac {a_1} b$
+By [[Ratios of Fractions in Lowest Terms]]:
+:$a_0 \divides a_1$
+where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
+From [[Divisibility of Elements in Geometric Sequence of Integers]]:
+:$a_0 \divides a_n$
+But $a_0 \perp a_n$.
+From this [[Proof by Contradiction|contradiction]] it follows that there can be no such $b$.
+{{qed}}
+{{Euclid Note|17|IX}}
+\end{proof}<|endoftext|>
+\section{Condition for Existence of Third Number Proportional to Two Numbers}
+Tags: Proportion
+
+\begin{theorem}
+Let $a, b, c \in \Z$ be [[Definition:Integer|integers]].
+Let $\tuple {a, b, c}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence]].
+In order for this to be possible, both of these conditions must be true:
+:$(1): \quad a$ and $b$ cannot be [[Definition:Coprime Integers|coprime]]
+:$(2): \quad a \divides b^2$
+where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
+{{:Euclid:Proposition/IX/18}}
+\end{theorem}
+
+\begin{proof}
+Let $P = \tuple {a, b, c}$ be a [[Definition:Geometric Sequence of Integers|geometric sequence]].
+Then by definition their [[Definition:Common Ratio|common ratio]] is:
+:$\dfrac b a = \dfrac c b$
+From [[Two Coprime Integers have no Third Integer Proportional]] it cannot be the case that $a$ and $b$ are [[Definition:Coprime Integers|coprime]].
+Thus condition $(1)$ is satisfied.
+From [[Form of Geometric Sequence of Integers]], $P$ is in the form:
+:$\tuple {k p^2, k p q, k q^2}$
+from which it can be seen that:
+:$k p^2 \divides k^2 p^2 q^2$
+demonstrating that condition $(2)$ is satisfied.
+{{qed}}
+{{Euclid Note|18|IX}}
+\end{proof}<|endoftext|>
+\section{Odd Divisor of Even Number also divides its Half}
+Tags: Euclidean Number Theory
+
+\begin{theorem}
+Let $a, b \in \Z$ be [[Definition:Integer|integers]].
+Let $a$ be [[Definition:Odd Integer|odd]] and $b$ be [[Definition:Even Integer|even]].
+Let:
+:$a \divides b$
+where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
+Then:
+:$a \divides \dfrac b 2$ 
+{{:Euclid:Proposition/IX/30}}
+\end{theorem}
+
+\begin{proof}
+By definition of an [[Definition:Even Number|even number]]:
+:$\exists r \in \Z: b = 2 r$
+By definition of an [[Definition:Odd Number|odd number]]:
+:$\exists s \in \Z: a = 2 s + 1$
+Thus:
+:$a \divides 2 \iff a = \pm 1$
+in which case from [[One Divides all Integers]]:
+:$a \divides \dfrac b 2$
+We have that:
+:$a \divides 2 r$
+and so by [[Euclid's Lemma]]:
+:$a \divides r = \dfrac b 2$
+{{qed}}
+{{Euclid Note|30|IX}}
+\end{proof}<|endoftext|>
+\section{Odd Number Coprime to Number is also Coprime to its Double}
+Tags: Euclidean Number Theory
+
+\begin{theorem}
+Let $a, b \in \Z$ be [[Definition:Integer|integers]].
+Let $a$ be [[Definition:Odd Integer|odd]].
+Let:
+: $a \perp b$
+where $\perp$ denotes [[Definition:Coprime Integers|coprimality]].
+Then:
+: $a \perp 2 b$
+{{:Euclid:Proposition/IX/31}}
+\end{theorem}
+
+\begin{proof}
+By definition of [[Definition:Odd Integer|odd number]]:
+:$a \perp 2$
+The result follows from [[Integer Coprime to all Factors is Coprime to Whole]].
+{{qed}}
+{{Euclid Note|31|IX}}
+\end{proof}<|endoftext|>
+\section{Power of Two is Even-Times Even Only}
+Tags: Euclidean Number Theory
+
+\begin{theorem}
+Let $a > 2$ be a [[Definition:Integer Power|power]] of $2$.
+Then $a$ is [[Definition:Even-Times Even Integer|even-times even]] only.
+{{:Euclid:Proposition/IX/32}}
+\end{theorem}
+
+\begin{proof}
+As $a$ is a [[Definition:Integer Power|power]] of $2$ greater than $2$:
+:$\exists k \in \Z_{>1}: a = 2^k$
+Thus:
+:$a = 2^2 2^{k - 2}$
+and so has $2^2 = 4$ as a [[Definition:Divisor of Integer|divisor]].
+Let $b$ be an [[Definition:Odd Number|odd number]].
+By definition:
+:$b \nmid 2$
+The result follows by [[Integer Coprime to all Factors is Coprime to Whole]].
+{{qed}}
+{{Euclid Note|32|IX}}
+\end{proof}<|endoftext|>
+\section{Number whose Half is Odd is Even-Times Odd}
+Tags: Euclidean Number Theory
+
+\begin{theorem}
+Let $a \in \Z$ be an [[Definition:Integer|integer]] such that $\dfrac a 2$ is an [[Definition:Odd Integer|odd integer]].
+Then $a$ is [[Definition:Even-Times Odd Integer|even-times odd]].
+{{:Euclid:Proposition/IX/33}}
+\end{theorem}
+
+\begin{proof}
+By definition:
+:$a = 2 r$
+where $r$ is an [[Definition:Odd Integer|odd integer]].
+Thus:
+:$a$ has an [[Definition:Even Integer|even]] [[Definition:Divisor of Integer|divisor]]
+and:
+:$a$ has an [[Definition:Odd Integer|odd]] [[Definition:Divisor of Integer|divisor]].
+Hence the result by definition of [[Definition:Even-Times Odd Integer|even-times odd integer]].
+As $r$ is an [[Definition:Odd Integer|odd integer]] it follows that $2 \nmid r$.
+Thus $a$ is not [[Definition:Divisor of Integer|divisible]] by $4$.
+Hence $a$ is not [[Definition:Even-Times Even Integer|even-times even]].
+{{qed}}
+{{Euclid Note|33|IX}}
+\end{proof}<|endoftext|>
+\section{Number neither whose Half is Odd nor Power of Two is both Even-Times Even and Even-Times Odd}
+Tags: Euclidean Number Theory
+
+\begin{theorem}
+Let $a \in \Z$ be an [[Definition:Integer|integer]] such that:
+:$(1): \quad a$ is not a [[Definition:Integer Power|power]] of $2$
+:$(2): \quad \dfrac a 2$ is an [[Definition:Even Integer|even integer]].
+Then $a$ is both [[Definition:Even-Times Even Integer|even-times even]] and [[Definition:Even-Times Odd Integer|even-times odd]].
+{{:Euclid:Proposition/IX/34}}
+\end{theorem}
+
+\begin{proof}
+As $\dfrac a 2$ is an [[Definition:Even Integer|even integer]] it follows that:
+:$\dfrac a 2 = 2 r$
+for some $r \in \Z$.
+That is:
+:$a = 2^2 r$
+and so $a$ is [[Definition:Even-Times Even Integer|even-times even]] by definition.
+{{qed}}
+Suppose $a$ is not [[Definition:Even-Times Odd Integer|even-times odd]].
+Then $a$ does not have an [[Definition:Odd Integer|odd]] [[Definition:Divisor of Integer|divisor]].
+Thus in its [[Definition:Prime Decomposition|prime decomposition]] there are no [[Definition:Odd Prime|odd primes]].
+Thus $a$ is in the form:
+:$a = 2^k$
+for some $k \in \Z_{>0}$.
+That is, $a$ is a [[Definition:Integer Power|power]] of $2$.
+This contradicts condition $(1)$.
+From this [[Proof by Contradiction|contradiction]] it is deduced that $a$ is [[Definition:Even-Times Odd Integer|even-times odd]].
+{{qed}}
+{{Euclid Note|34|IX}}
+\end{proof}<|endoftext|>
+\section{Scaled Real Function that Increases Without Bound}
+Tags: Unbounded Mappings
+
+\begin{theorem}
+Let $f: \R \to \R$ be a [[Definition:Real Function|real function]].
+Let $\lambda \in \R_{\ne 0}$ be a nonzero [[Definition:Constant|constant]].
+Then:
+For $\lambda > 0$:
+:$\displaystyle \lim_{x \mathop \to +\infty} \map f x = +\infty \implies \lim_{x \mathop \to +\infty} \lambda \map f x = +\infty$
+:$\displaystyle \lim_{x \mathop \to -\infty} \map f x = +\infty \implies \lim_{x \mathop \to -\infty} \lambda \map f x = + \infty$
+For $\lambda < 0$:
+:$\displaystyle \lim_{x \mathop \to +\infty} \map f x = +\infty \implies \lim_{x \mathop \to +\infty} \lambda \map f x = -\infty$
+:$\displaystyle \lim_{x \mathop \to -\infty} \map f x = +\infty \implies \lim_{x \mathop \to -\infty} \lambda \map f x = -\infty$
+\end{theorem}
+
+\begin{proof}
+Let $\displaystyle \lim_{x \mathop \to +\infty} \map f x = +\infty$.
+From the definition of [[Definition:Infinite Limit at Infinity|infinite limit at infinity]], this means that:
+:$\forall M > 0: \exists N > 0: x > N \implies \map f x > M$.
+Suppose $\lambda > 0$.
+Then $M > 0 \iff \lambda^{-1} M > 0$.
+Also, $\map f x > \lambda^{-1} M \iff \lambda f\map f x > M$.
+So:
+:$\forall M > 0: \exists N > 0: x > N \implies \lambda \map f x > M$
+From the definition of [[Definition:Infinite Limit at Infinity|infinite limit at infinity]]:
+:$\displaystyle \lim_{x \mathop \to +\infty} \lambda \map f x = +\infty$.
+The proof for $\displaystyle \lim_{x \mathop \to -\infty} \map f x = +\infty$ is analogous.
+Now, suppose $\lambda < 0$.
+Then $-\lambda f > 0$, and so $-\lambda f \to + \infty$, from above.
+Write $\lambda f = -\paren {-\lambda f}$ and the result follows from [[Negative of Real Function that Increases Without Bound]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Scaled Real Function that Decreases Without Bound}
+Tags: Unbounded Mappings
+
+\begin{theorem}
+Let $f: \R \to \R$ be a [[Definition:Real Function|real function]].
+Let $\lambda \in \R_{\ne 0}$ be a nonzero [[Definition:Constant|constant]].
+Then:
+For $\lambda > 0$:
+:$\displaystyle \lim_{x \mathop \to +\infty} \map f x = -\infty \implies \lim_{x \mathop \to +\infty} \lambda \map f x = -\infty$
+:$\displaystyle \lim_{x \mathop \to -\infty} \map f x = -\infty \implies \lim_{x \mathop \to -\infty} \lambda \map f x = -\infty$
+For $\lambda < 0$:
+:$\displaystyle \lim_{x \mathop \to +\infty} \map f x = -\infty \implies \lim_{x \mathop \to +\infty} \lambda \map f x = +\infty$
+:$\displaystyle \lim_{x \mathop \to -\infty} \map f x = -\infty \implies \lim_{x \mathop \to -\infty} \lambda \map f x = +\infty$
+\end{theorem}
+
+\begin{proof}
+Let $\displaystyle \lim_{x \mathop \to +\infty} \map f x = -\infty$.
+From the definition of [[Definition:Infinite Limit at Infinity|infinite limits at infinity]], this means that:
+:$\forall M < 0: \exists N > 0: x > N \implies \map f x < M$
+Suppose $\lambda > 0$.
+Then $M < 0 \iff \lambda^{-1}M < 0$.
+Also, $\map f x < \lambda^{-1}M \iff \lambda \map f x < M$
+So:
+:$\forall M < 0: \exists N > 0: x > N \implies \lambda \map f x < M$
+From the definition of [[Definition:Infinite Limit at Infinity|infinite limits at infinity]]:
+:$\displaystyle \lim_{x \mathop \to +\infty} \lambda \map f x = -\infty$
+The proof for $\displaystyle  \lim_{x \mathop \to -\infty} \map f x = -\infty$ is analogous.
+Now, suppose $\lambda < 0$.
+Then $-\lambda f < 0$, and so $-\lambda f \to -\infty$, from above.
+Write $\lambda f = -\paren {-\lambda f}$ and the result follows from [[Negative of Real Function that Decreases Without Bound]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Existence of Fraction of Number Smaller than Given}
+Tags: Euclidean Number Theory
+
+\begin{theorem}
+Let $a, b \in \R_{>0}$ be two [[Definition:Strictly Positive Real Number|(strictly) positive real numbers]], such that $a > b$.
+Let the [[Definition:Sequence|sequence]] $\left\langle{a_n}\right\rangle$ be defined [[Definition:Recursive Sequence|recursively]] as:
+:$a_i = \begin{cases} a & : i = 1 \\
+a_{i - 1} - c: \dfrac {a_{i - 1} } 2 < c < a_{i - 1} & : i > 1
+\end{cases}$
+Then:
+: $\exists n \in \N_{>0}: a_n < b$
+{{:Euclid:Proposition/X/1}}
+\end{theorem}
+
+\begin{proof}
+:[[File:Euclid-X-1.png|400px]]
+Let $AB$ and $C$ be two unequal [[Definition:Strictly Positive Real Number|magnitudes]] such that $AB > C$.
+From {{EuclidDefLink|V|4|Existence of Ratio}} there exists a [[Definition:Multiple|multiple]] of $C$ which is greater than $AB$.
+Thus let $DE$ be a [[Definition:Multiple|multiple]] of $C$ which is greater than $AB$.
+Let $DE$ be divided into parts $DF$, $FG$ and $GE$ (for example) which are equal to $C$.
+From $AB$ let $BH$ be subtracted which is greater than half $AB$.
+From $AH$ let $HK$ be subtracted which is greater than half $AH$.
+Let this process be continued until the divisions of $AB$ are equal in number to the divisions of $DE$.
+Let $AK, KH, HB$ be divisions which are equal in number to $DF, FG, GE$.
+We have that $DE > AB$.
+From $DE$ there has been subtracted $EG$ which is less than half $DE$.
+From $AB$ there has been subtracted $BH$ which is greater than half $AB$.
+Therefore the remainder $GD$ of $DE$ is greater than the remainder $HA$ of $AB$.
+From $GD$ there has been subtracted $GF$ which is half $GD$.
+From $HA$ there has been subtracted $HK$ which is greater than half $HA$.
+Therefore the remainder $DF$ of $GD$ is greater than the remainder $AK$ of $HA$.
+But $DF = C$.
+Therefore $C > AK$.
+Therefore there is left of the [[Definition:Strictly Positive Real Number|magnitude]] $AB$ the [[Definition:Strictly Positive Real Number|magnitude]] $AK$ which is less than the lesser [[Definition:Strictly Positive Real Number|magnitude]] set out, which is $C$.
+{{qed}}
+{{Euclid Note|1|X}}
+\end{proof}<|endoftext|>
+\section{Restriction of Continuous Mapping is Continuous/Topological Spaces}
+Tags: Topology, Continuous Mappings
+
+\begin{theorem}
+Let $T_1 = \left({S_1, \tau_1}\right)$ and $T_2 = \left({S_2, \tau_2}\right)$ be [[Definition:Topological Space|topological spaces]].
+Let $M_1 \subseteq S_1$ be a [[Definition:Subset|subset]] of $S_1$.
+Let $f: S_1 \to S_2$ be a [[Definition:Mapping|mapping]] which is [[Definition:Continuous Mapping (Topological Spaces)|continuous]].
+Let $M_2 \subseteq S_2$ be a [[Definition:Subset|subset]] of $S_2$ such that $f\left[{M_1}\right] \subseteq M_2$.
+Let $f {\restriction_{M_1 \times M_2}}: M_1 \to M_2$ be the [[Definition:Restriction of Mapping |restriction]] of $f$ to $M_1 \times M_2$.
+Then $f {\restriction_{M_1 \times M_2}}$ is [[Definition:Continuous Mapping (Topological Spaces)|continuous]], where $M_1$ and $M_2$ are equipped with the respective [[Definition:Subspace Topology|subspace topologies]].
+\end{theorem}
+
+\begin{proof}
+Let $V \subseteq M_2$ be an [[Definition:Open Set (Topology)|open set]] with respect to the [[Definition:Subspace Topology|subspace topology]] of $M_2$.
+By definition of the [[Definition:Subspace Topology|subspace topology]], $V = U \cap M_2$ for an [[Definition:Open Set (Topology)|open set]] $U \in \tau_2$.
+We have that:
+{{begin-eqn}}
+{{eqn | l = \left({f {\restriction_{M_1 \times M_2} } }\right)^{-1} \left[{V}\right]
+      | r = \left\{ {x \in M_1 : f {\restriction_{M_1 \times M_2} } \left({x}\right) \in V} \right\}
+      | c = Definition of [[Definition:Preimage of Subset under Mapping|Preimage]]
+}}
+{{eqn | r = \left\{ {x \in M_1 : f {\restriction_{M_1 \times M_2} } \left({x}\right) \in U}\right\}
+      | c = $f \left({M_1}\right) \subseteq M_2$
+}}
+{{eqn | r = \left\{ {x \in M_1 : f \left({x}\right) \in U}\right\}
+      | c = Definition of [[Definition:Restriction of Mapping|Restriction of Mapping]]
+}}
+{{eqn | r = \left\{ {x \in S_1 : f \left({x}\right) \in U}\right\} \cap M_1
+      | c = Definition of [[Definition:Set Intersection|Set Intersection]]
+}}
+{{eqn | r = f^{-1} \left[{U}\right] \cap M_1
+      | c = Definition of [[Definition:Preimage of Subset under Mapping|Preimage]]
+}}
+{{end-eqn}}
+By definition of [[Definition:Continuous Mapping (Topological Spaces)|continuous mapping]], $f^{-1} \left[{U}\right]$ is [[Definition:Open Set (Topology)|open]] in $T_1$.
+By definition of [[Definition:Subspace Topology|subspace topology]], $f^{-1} \left[{U}\right] \cap M_1$ is [[Definition:Open Set (Topology)|open]] with respect to the [[Definition:Subspace Topology|subspace topology]] of $M_1$.
+As $f^{-1} \left[{U}\right] \cap M_1 = {f {\restriction_{M_1 \times M_2}}}^{-1} \left[{V}\right]$ it follows that ${f {\restriction_{M_1 \times M_2}}}^{-1} \left[{V}\right]$ is [[Definition:Open Set (Topology)|open]] with respect to the [[Definition:Subspace Topology|subspace topology]] of $M_1$.
+Since $V \subseteq M_2$ was an arbitrary [[Definition:Open Set (Topology)|open set]], $f$ is [[Definition:Continuous Mapping (Topological Spaces)|continuous]].
+{{qed}}
+[[Category:Topology]]
+[[Category:Continuous Mappings]]
+s06ioeumwuwzpcvqj6g9ux7ciieh01e
+\end{proof}<|endoftext|>
+\section{Restriction of Continuous Mapping is Continuous/Metric Spaces}
+Tags: Metric Spaces, Continuous Mappings on Metric Spaces
+
+\begin{theorem}
+Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be [[Definition:Metric Space|metric spaces]].
+Let $S \subseteq M_1$ be a [[Definition:Subset|subset]] of $M_1$.
+Let $f: M_1 \to M_2$ be a [[Definition:Mapping|mapping]] which is [[Definition:Continuous Mapping (Metric Spaces)|continuous]] at a point $\alpha \in S$.
+Let $f \restriction_S = g: S \to M_2$ be the [[Definition:Restriction of Mapping |restriction]] of $f$ to $S$.
+Then $g$ is [[Definition:Continuous Mapping (Metric Spaces)|continuous]] at $\alpha$.
+\end{theorem}
+
+\begin{proof}
+Let $\sequence {z_n}$ be a [[Definition:Sequence|sequence in $S$]] such that:
+:$\displaystyle \lim_{n \mathop \to \infty} z_n = \alpha$
+Since $\sequence {z_n}$ and $\alpha$ both lie in $S$:
+:$\displaystyle \lim_{n \mathop \to \infty} \map f {z_n} \to \alpha$
+But:
+:$\forall n \in \N: \map g {z_n} = \map f {z_n}$
+and also:
+:$\map g \alpha = \map f \alpha$
+The result follows from [[Limit of Function by Convergent Sequences]].
+{{qed}}
+[[Category:Metric Spaces]]
+[[Category:Continuous Mappings on Metric Spaces]]
+dznt78ygvoyxfoho264sgdkxf28je6l
+\end{proof}<|endoftext|>
+\section{Restriction of Inverse is Inverse of Restriction}
+Tags: Restrictions, Inverse Mappings
+
+\begin{theorem}
+Let $S_1$ and $S_2$ be [[Definition:Set|sets]].
+Let $f: S_1 \to S_2$ be a [[Definition:Bijection|bijection]].
+Let $S \subseteq S_1$ be a [[Definition:Subset|subset]] of $S_1$.
+Let $f^{-1}$ be the [[Definition:Inverse Mapping|inverse]] of $f$.
+Let $f {\restriction_{S \times f \left[{S}\right]}}$ be the [[Definition:Restriction of Mapping|restriction]] of $f$ to $S \times f\left[{S}\right]$.
+Let $f^{-1} {\restriction_{f \left[{S}\right] \times S}}$ be the [[Definition:Restriction of Mapping|restriction]] of $f^{-1}$ to $f \left[{S}\right] \times S$.
+Then:
+: $f {\restriction_{S \times f \left[{S}\right]}}$ is a [[Definition:Bijection|bijection]]
+and:
+: ${\left({f {\restriction_{S \times f \left[{S}\right]}} }\right)}^{-1} = f^{-1} {\restriction_{f \left[{S}\right] \times S}}$
+\end{theorem}
+
+\begin{proof}
+Let $y \in f \left[{S}\right]$.
+By the definition of [[Definition:Image of Subset under Mapping|image]]:
+:$\exists z \in S : f \left({z}\right) = y$
+Since $f$ is a [[Definition:Bijection|bijection]], $f^{-1}$ is a [[Definition:Mapping|mapping]].
+Let $x = f^{-1} \left({y}\right)$.
+Suppose that $x \notin S$.
+Suppose that $x = z$.
+Then $x \in S$.
+Thus $x \ne z$.
+By the definition of [[Definition:Inverse Mapping|inverse mapping]]:
+: $f\left({x}\right) = y$
+By [[Equality is Transitive]]:
+: $f \left({x}\right) = f \left({z}\right)$
+Thus, $f$ is not an [[Definition:Injection|injection]].
+Thus, $f$ is not a [[Definition:Bijection|bijection]].
+This contradicts the assumption.
+Therefore, $x \in S$.
+By the definition of [[Definition:Restriction of Mapping|restriction of mapping]]:
+: $f {\restriction_{S \times f \left[{S}\right]}} \left({x}\right) = f\left({x}\right)$
+By [[Equality is Transitive]]:
+: $f {\restriction_{S \times f \left[{S}\right]}} \left({x}\right) = y$
+Suppose that there exists a $w \in S$ such that $w \ne x$ and $f {\restriction_{S \times f \left[{S}\right]}} \left({w}\right) = y$.
+By the definition of [[Definition:Restriction of Mapping|restriction of mapping]]:
+: $f {\restriction_{S \times f \left[{S}\right]}} \left({w}\right) = f\left({w}\right)$
+By [[Equality is Transitive]]:
+: $f \left({w}\right) = y$
+By [[Equality is Transitive]]:
+: $f \left({w}\right) = f\left({x}\right)$
+Thus, $f$ is not an [[Definition:Injection|injection]].
+Thus, $f$ is not a [[Definition:Bijection|bijection]].
+This contradicts the assumption.
+Therefore, there exists a [[Definition:Unique|unique]] $x \in S$ such that:
+: $f {\restriction_{S \times f\left[{S}\right]}} \left({x}\right) = y$
+Since $y \in f\left[{S}\right]$ was arbitrary, $f {\restriction_{S \times f \left[{S}\right]}}$ is a [[Definition:Bijection|bijection]].
+Thus, ${\left(f {\restriction_{S \times f \left[{S}\right]}}\right)}^{-1}$ is a [[Definition:Mapping|mapping]].
+Let $x' = {\left(f {\restriction_{S \times f \left[{S}\right]}}\right)}^{-1}\left({y}\right)$.
+By the definition of [[Definition:Inverse Mapping|inverse mapping]]:
+: $f {\restriction_{S \times f \left[{S}\right]}} \left({x'}\right) = y$
+By the definition of [[Definition:Restriction of Mapping|restriction of mapping]], we have $f \left({x'}\right) = y$.
+Suppose that $x' \ne x$.
+By [[Equality is Transitive]]:
+: $f\left({x'}\right) = f\left({x}\right)$
+Thus, $f$ is not an [[Definition:Injection|injection]].
+Thus, $f$ is not a [[Definition:Bijection|bijection]].
+This contradicts the assumption.
+Therefore, $x' = x$.
+By [[Equality is Transitive]]:
+: $f^{-1}\left({y}\right) = (f {\restriction_{S \times f \left[{S}\right]}})^{-1} \left({y}\right)$
+By the definition of [[Definition:Restriction of Mapping|restriction of mapping]]:
+: $f^{-1} {\restriction_{f \left[{S}\right] \times S}} \left({y}\right) = f^{-1} \left({y}\right)$
+By [[Equality is Transitive]]:
+: ${\left(f {\restriction_{S \times f \left[{S}\right]}}\right)}^{-1} \left({y}\right) = f^{-1} {\restriction_{f\left[{S}\right] \times S}} \left({y}\right)$
+Since $y \in f\left[{S}\right]$ was arbitrary:
+: ${\left(f {\restriction_{S \times f \left[{S}\right]}}\right)}^{-1} = f^{-1} {\restriction_{f \left[{S}\right] \times S}}$
+{{qed}}
+[[Category:Restrictions]]
+[[Category:Inverse Mappings]]
+ge779b0r8bkxb2vrgidb7cpyz5v4qid
+\end{proof}<|endoftext|>
+\section{Restriction of Homeomorphism is Homeomorphism}
+Tags: Homeomorphisms
+
+\begin{theorem}
+Let $T_1 = \left({S_1, \tau_1}\right)$, $T_2 = \left({S_2, \tau_2}\right)$ be [[Definition:Topological Space|topological spaces]].
+Let $f: S_1 \to S_2$ be a [[Definition:Homeomorphism|homeomorphism]] between $T_1$ and $T_2$.
+Let $S$ be a [[Definition:Subset|subset]] of $S_1$.
+Let $f {\restriction_{S \times f\left[{S}\right]}} : S \to f \left[{S}\right]$ be the [[Definition:Restriction of Mapping|restriction]] of $f$ to $S \times f \left[{S}\right]$.
+Let $S$ and $f \left[{S}\right]$ bear their respective [[Definition:Subspace Topology|subspace topologies]].
+Then $f {\restriction_{S \times f \left[{S}\right]}}$ is a [[Definition:Homeomorphism|homeomorphism]].
+\end{theorem}
+
+\begin{proof}
+By [[Restriction of Continuous Mapping is Continuous/Topological Spaces|Restriction of Continuous Mapping is Continuous]], $f {\restriction_{S \times f \left[{S}\right]}}$ is [[Definition:Continuous Mapping on Set|continuous]].
+By [[Restriction of Inverse is Inverse of Restriction]], $\left({f {\restriction_{S \times f \left[{S}\right]}}}\right)^{-1}$ is [[Definition:Well-Defined Mapping|well-defined]] and equal to $f^{-1} {\restriction_{f \left[{S}\right] \times S}}$.
+By [[Restriction of Continuous Mapping is Continuous/Topological Spaces|Restriction of Continuous Mapping is Continuous]], $f^{-1} {\restriction_{f \left[{S}\right] \times S}}$ is [[Definition:Continuous Mapping on Set|continuous]].
+Since $\left({f {\restriction_{S \times f \left[{S}\right]}}}\right)^{-1} = f^{-1} {\restriction_{f \left[{S}\right] \times S}}$, $\left({f {\restriction_{S \times f \left[{S}\right]}}}\right)^{-1}$ is [[Definition:Continuous Mapping on Set|continuous]].
+Hence, $f {\restriction_{S \times f \left[{S}\right]}}$ is a [[Definition:Homeomorphism|homeomorphism]].
+{{qed}}
+[[Category:Homeomorphisms]]
+99a6bp3gqgbkt7cucgrah6bato7al5k
+\end{proof}<|endoftext|>
+\section{Hölder's Inequality/General}
+Tags: Lebesgue Spaces
+
+\begin{theorem}
+Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
+For $i = 1, \ldots, n$ let $p_i \in \R$ such that:
+:$\displaystyle \sum_{i \mathop = 1}^n \frac 1 {p_i} = 1$
+Let $f_i \in \map {\LL^{p_i} } \mu, f_i: X \to \R$, where $\LL$ denotes [[Definition:Lebesgue Space|Lebesgue space]].
+Then their [[Definition:Pointwise Product|pointwise product]] $\displaystyle \prod_{i \mathop = 1}^n f_i$ is [[Definition:Integrable Function on Measure Space|integrable]], that is:
+:$\displaystyle \prod_{i \mathop = 1}^n f_i \in \map {\LL^1} \mu$
+and:
+:$\displaystyle \norm {\prod_{i \mathop = 1}^n f_i}_1 = \int \size {\prod_{i \mathop = 1}^n f_i} \rd \mu \le \prod_{i \mathop = 1}^n \norm {f_i}_{p_i}$
+where the various instances of $\norm {\, \cdot \,}$ signify [[Definition:P-Seminorm|$p$-seminorms]].
+\end{theorem}
+
+\begin{proof}
+{{explain|Has it been demonstrated for $i {{=}} 1$? Best if this were put into the house style of induction proofs.}}
+We use the [[Principle of Mathematical Induction]].
+Let it be assumed that the result holds for $i = n - 1$.
+We show that the result holds for $i = n$. 
+Define:
+:$q_n := \dfrac {p_n} {p_n-1}$
+and for $i = 1, \ldots, n - 1$, define:
+:$r_i := p_i \cdot \paren {1 - \dfrac 1 {p_n} }$
+Then:
+:$\dfrac 1 {p_n} + \dfrac 1 {q_n} = 1$
+:$\displaystyle \sum_{i \mathop = 1}^{n-1} \dfrac 1 {r_i} = 1$
+and:
+:$q_n \cdot r_i = p_i$
+Applying [[Hölder's Inequality]] to $\displaystyle f := \prod_{i \mathop = 1}^{n-1} f_i$ and $g := f_n$, we find:
+{{begin-eqn}}
+{{eqn | l = \int \size {\prod_{i \mathop = 1}^n f_i} \rd \mu
+      | o = \le
+      | r = \norm {f_n}_{p_n} \cdot \norm {\prod_{i \mathop = 1}^{n - 1} f_i}_{q_n}
+      | c = 
+}}
+{{eqn | r = \norm {f_n}_{p_n} \cdot \paren {\int \prod_{i \mathop = 1}^{n - 1} \size { {f_i}^{q_n} } \rd \mu}^\frac 1 {q_n}
+      | c = 
+}}
+{{eqn | o = \le
+      | r = \norm {f_n}_{p_n} \cdot \paren {\prod_{i \mathop = 1}^{n-1} \norm { {f_i}^{q_n} }_{r_i} }^\frac 1 {q_n}
+      | c = 
+}}
+{{eqn | r = \prod_{i \mathop = 1}^n \norm {f_i}_{p_i}
+      | c = 
+}}
+{{end-eqn}}
+{{qed}}
+{{Namedfor|Otto Ludwig Hölder}}
+[[Category:Lebesgue Spaces]]
+pgq1ogkfb0jnvkjt0g7kkhxhtwzcmtz
+\end{proof}<|endoftext|>
+\section{Theorem of Even Perfect Numbers/Sufficient Condition}
+Tags: Theorem of Even Perfect Numbers
+
+\begin{theorem}
+Let $n \in \N$ be such that $2^n - 1$ is [[Definition:Prime Number|prime]].
+Then $2^{n - 1} \paren {2^n - 1}$ is [[Definition:Perfect Number|perfect]].
+{{:Euclid:Proposition/IX/36}}
+\end{theorem}
+
+\begin{proof}
+Suppose $2^n - 1$ is [[Definition:Prime Number|prime]].
+Let $a = 2^{n - 1} \paren {2^n - 1}$.
+Then $n \ge 2$ which means $2^{n - 1}$ is [[Definition:Even Integer|even]] and hence so is $a = 2^{n - 1} \paren {2^n - 1}$.
+Note that $2^n - 1$ is [[Definition:Odd Integer|odd]].
+Since all [[Definition:Divisor of Integer|divisors]] (except $1$) of $2^{n - 1}$ are [[Definition:Even Integer|even]] it follows that $2^{n - 1}$ and $2^n - 1$ are [[Definition:Coprime Integers|coprime]].
+Let $\map \sigma n$ be the [[Definition:Sigma Function|sigma function]] of $n$, that is, the sum of all [[Definition:Divisor of Integer|divisors]] of $n$ (including $n$).
+From [[Sigma Function is Multiplicative]], it follows that $\map \sigma a = \map \sigma {2^{n - 1} } \map \sigma {2^n - 1}$.
+But as $2^n - 1$ is prime, $\map \sigma {2^n - 1} = 2^n$ from [[Sigma Function of Prime Number]].
+Then we have that $\map \sigma {2^{n - 1} } = 2^n - 1$ from [[Sigma Function of Power of Prime]].
+Hence it follows that $\map \sigma a = \paren {2^n - 1} 2^n = 2 a$.
+Hence from the definition of [[Definition:Perfect Number|perfect number]] it follows that $2^{n - 1} \paren {2^n - 1}$ is [[Definition:Perfect Number|perfect]].
+\end{proof}<|endoftext|>
+\section{Theorem of Even Perfect Numbers/Necessary Condition}
+Tags: Theorem of Even Perfect Numbers
+
+\begin{theorem}
+Let $a \in \N$ be an [[Definition:Even Integer|even]] [[Definition:Perfect Number|perfect number]].
+Then $a$ is in the form:
+:$2^{n - 1} \paren {2^n - 1}$
+where $2^n - 1$ is [[Definition:Prime Number|prime]].
+\end{theorem}
+
+\begin{proof}
+Let $a \in \N$ be an [[Definition:Even Integer|even]] [[Definition:Perfect Number|perfect number]].
+We can extract the highest power of $2$ out of $a$ that we can, and write $a$ in the form:
+:$a = m 2^{n-1}$
+where $n \ge 2$ and $m$ is [[Definition:Odd Integer|odd]].
+Since $a$ is [[Definition:Perfect Number|perfect]] and therefore $\map \sigma a = 2 a$:
+{{begin-eqn}}     
+{{eqn| l = m 2^n
+     | r = 2 a
+     | c =
+}}
+{{eqn| r = \map \sigma a
+     | c =
+}}
+{{eqn| r = \map \sigma {m 2^{n - 1} }
+     | c =
+}}
+{{eqn| r = \map \sigma m \map \sigma {2^{n - 1} }
+     | c = [[Sigma Function is Multiplicative]]
+}}
+{{eqn| r = \map \sigma m {2^n - 1}
+     | c = [[Sigma Function of Power of Prime]]
+}}
+{{end-eqn}}
+So:
+:$\map \sigma m = \dfrac {m 2^n} {2^n - 1}$
+But $\map \sigma m$ is an [[Definition:Integer|integer]] and so $2^n - 1$ [[Definition:Divisor of Integer|divides]] $m 2^n$.
+From [[Consecutive Integers are Coprime]], $2^n$ and $2^n - 1$ are [[Definition:Coprime Integers|coprime]].
+So from [[Euclid's Lemma]] $2^n - 1$ [[Definition:Divisor of Integer|divides]] $m$.
+Thus $\dfrac m {2^n - 1}$ [[Definition:Divisor of Integer|divides]] $m$.
+Since $2^n - 1 \ge 3$ it follows that:
+:$\dfrac m {2^n - 1} < m$
+Now we can express $\map \sigma m$ as:
+:$\map \sigma m = \dfrac {m 2^n} {2^n - 1} = m + \dfrac m {2^n - 1}$
+This means that the sum of all the [[Definition:Divisor of Integer|divisors]] of $m$ is equal to $m$ itself plus one other divisor of $m$.
+Hence $m$ must have exactly two [[Definition:Divisor of Integer|divisors]], so it must be [[Definition:Prime Number|prime]] by definition.
+This means that the other [[Definition:Divisor of Integer|divisor]] of $m$, apart from $m$ itself, must be $1$.
+That is:
+:$\dfrac m {2^n - 1} = 1$
+Hence the result.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Condition for Commensurability of Roots of Quadratic Equation/Lemma}
+Tags: Euclidean Number Theory
+
+\begin{theorem}
+{{:Euclid:Proposition/X/17/Lemma}}
+\end{theorem}
+
+\begin{proof}
+:[[File:Euclid-X-17-Lemma.png|300px]]
+Let $AB$ be a [[Definition:Line Segment|straight line]].
+Let the [[Definition:Parallelogram|parallelogram]] $AD$ be applied to $AB$ which is deficient by the [[Definition:Square (Geometry)|square]] $DB$.
+As $DB$ is a [[Definition:Square (Geometry)|square]]:
+: $DC = BC$
+Thus $AD$ is the [[Definition:Rectangle|rectangle]] [[Definition:Containment of Rectangle|contained]] by $AC$ and $CB$.
+{{qed}}
+{{Euclid Note|17|X}}
+\end{proof}<|endoftext|>
+\section{Condition for Commensurability of Roots of Quadratic Equation}
+Tags: Euclidean Number Theory
+
+\begin{theorem}
+Consider the [[Definition:Quadratic Equation|quadratic equation]]:
+:$(1): \quad a x - x^2 = \dfrac {b^2} 4$
+Then $x$ and $a - x$ are [[Definition:Commensurable|commensurable]] {{iff}} $\sqrt{a^2 - b^2}$ and $a$ are [[Definition:Commensurable|commensurable]].
+{{:Euclid:Proposition/X/17}}
+\end{theorem}
+
+\begin{proof}
+We have that:
+{{begin-eqn}}
+{{eqn | l = x \left({a - x}\right) + \left({\frac a 2 - x}\right)^2
+      | r = a x - x^2 + \frac {a^2} 4 - 2 \frac a 2 x + x^2
+      | c = 
+}}
+{{eqn | r = \frac {a^2} 4
+      | c = simplifying
+}}
+{{eqn | ll= \implies
+      | l = 4 x \left({a - x}\right) + 4 \left({\frac a 2 - x}\right)^2
+      | r = a^2
+      | c = 
+}}
+{{eqn | ll= \implies
+      | l = b^2 + \left({a - 2 x}\right)^2
+      | r = a^2
+      | c = from $(1)$
+}}
+{{eqn | ll= \implies
+      | l = a^2 - b^2
+      | r = \left({a - 2 x}\right)^2
+      | c = 
+}}
+{{eqn | ll= \implies
+      | l = \sqrt {a^2 - b^2}
+      | r = a - 2 x
+      | c = 
+}}
+{{end-eqn}}
+Let $a \frown b$ denote that $a$ is [[Definition:Commensurable|commensurable]] with $b$.
+=== Necessary Condition ===
+Let $\left({a - x}\right) \frown x$.
+From [[Commensurability of Sum of Commensurable Magnitudes]]:
+:$a \frown x$
+From [[Magnitudes with Rational Ratio are Commensurable]]:
+:$x \frown 2 x$
+So:
+{{begin-eqn}}
+{{eqn | l = a
+      | o = \frown
+      | r = 2 x
+      | c = [[Commensurability is Transitive Relation]]
+}}
+{{eqn | o = \frown
+      | r = \left({a - 2 x}\right)
+      | c = [[Commensurability of Sum of Commensurable Magnitudes]]
+}}
+{{eqn | o = \frown
+      | r = \sqrt {a^2 - b^2}
+      | c = 
+}}
+{{end-eqn}}
+{{qed|lemma}}
+=== Sufficient Condition ===
+Let $a \frown \sqrt {a^2 - b^2}$.
+Then:
+{{begin-eqn}}
+{{eqn | l = a
+      | o = \frown
+      | r = \sqrt {a^2 - b^2}
+      | c = 
+}}
+{{eqn | o = \frown
+      | r = \left({a - 2 x}\right)
+      | c = 
+}}
+{{eqn | o = \frown
+      | r = 2 x
+      | c = [[Commensurability of Sum of Commensurable Magnitudes]]
+}}
+{{end-eqn}}
+From [[Magnitudes with Rational Ratio are Commensurable]]:
+:$x \frown 2 x$
+From [[Commensurability is Transitive Relation]]:
+:$a \frown x$
+From [[Commensurability of Sum of Commensurable Magnitudes]]:
+: $\left({a - x}\right) \frown x$
+{{qed}}
+{{Euclid Note|17|X}}
+\end{proof}<|endoftext|>
+\section{Condition for Incommensurability of Roots of Quadratic Equation}
+Tags: Euclidean Number Theory
+
+\begin{theorem}
+Consider the [[Definition:Quadratic Equation|quadratic equation]]:
+:$(1): \quad a x - x^2 = \dfrac {b^2} 4$
+Then $x$ and $a - x$ are [[Definition:Incommensurable|incommensurable]] {{iff}} $\sqrt {a^2 - b^2}$ and $a$ are [[Definition:Incommensurable|incommensurable]].
+{{:Euclid:Proposition/X/18}}
+\end{theorem}
+
+\begin{proof}
+We have that:
+{{begin-eqn}}
+{{eqn | l = x \paren {a - x} + \paren {\frac a 2 - x}^2
+      | r = a x - x^2 + \frac {a^2} 4 - 2 \frac a 2 x + x^2
+      | c = 
+}}
+{{eqn | r = \frac {a^2} 4
+      | c = simplifying
+}}
+{{eqn | ll= \leadsto
+      | l = 4 x \paren {a - x} + 4 \paren {\frac a 2 - x}^2
+      | r = a^2
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = b^2 + \paren {a - 2 x}^2
+      | r = a^2
+      | c = from $(1)$
+}}
+{{eqn | ll= \leadsto
+      | l = a^2 - b^2
+      | r = \paren {a - 2 x}^2
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = \sqrt {a^2 - b^2}
+      | r = a - 2 x
+      | c = 
+}}
+{{end-eqn}}
+Let:
+:$a \smile b$ denote that $a$ is [[Definition:Incommensurable|incommensurable]] with $b$
+:$a \frown b$ denote that $a$ is [[Definition:Commensurable|commensurable]] with $b$.
+=== Necessary Condition ===
+Let $\paren {a - x} \smile x$.
+From [[Incommensurability of Sum of Incommensurable Magnitudes]]:
+:$a \smile x$
+From [[Magnitudes with Rational Ratio are Commensurable]]:
+:$x \frown 2 x$
+So:
+{{begin-eqn}}
+{{eqn | l = a
+      | o = \smile
+      | r = 2 x
+      | c = [[Commensurable Magnitudes are Incommensurable with Same Magnitude]]
+}}
+{{eqn | o = \smile
+      | r = \paren {a - 2 x}
+      | c = [[Incommensurability of Sum of Incommensurable Magnitudes]]
+}}
+{{eqn | o = \smile
+      | r = \sqrt {a^2 - b^2}
+      | c = 
+}}
+{{end-eqn}}
+{{qed|lemma}}
+=== Sufficient Condition ===
+Let $a \smile \sqrt {a^2 - b^2}$.
+Then:
+{{begin-eqn}}
+{{eqn | l = a
+      | o = \smile
+      | r = \sqrt {a^2 - b^2}
+      | c = 
+}}
+{{eqn | o = \smile
+      | r = \paren {a - 2 x}
+      | c = 
+}}
+{{eqn | o = \smile
+      | r = 2 x
+      | c = [[Incommensurability of Sum of Incommensurable Magnitudes]]
+}}
+{{end-eqn}}
+From [[Magnitudes with Rational Ratio are Commensurable]]:
+:$x \frown 2 x$
+From [[Commensurable Magnitudes are Incommensurable with Same Magnitude]]:
+:$a \smile x$
+From [[Incommensurability of Sum of Incommensurable Magnitudes]]:
+:$\paren {a - x} \smile x$
+{{qed}}
+{{Euclid Note|18|X}}
+\end{proof}<|endoftext|>
+\section{Product of Rationally Expressible Numbers is Rational/Lemma}
+Tags: Euclidean Number Theory
+
+\begin{theorem}
+{{:Euclid:Proposition/X/19/Lemma}}
+\end{theorem}
+
+\begin{proof}
+This lemma does little more than restate (in a considerably more wordy form) {{EuclidPropLink|book = X|prop = 9|title = Commensurability of Squares}}.
+{{qed}}
+{{Euclid Note|19|X|It was suggested by {{AuthorRef|Johan Ludvig Heiberg|Heiberg}} that this [[Definition:Lemma|lemma]] is a later interpolation, and not part of the original work by {{AuthorRef|Euclid}}. In his own edition of this work, he relegated it to the appendix.<br/><br/>{{AuthorRef|Sir Thomas L. Heath|Heath}} similarly separates it out from the canon in his case by enclosing it in brackets $[\,]$.}}
+\end{proof}<|endoftext|>
+\section{Product of Rationally Expressible Numbers is Rational}
+Tags: Euclidean Number Theory
+
+\begin{theorem}
+Let $a, b \in \set {x \in \R_{>0} : x^2 \in \Q}$ be the lengths of [[Definition:Rational Line Segment|rational line segments]].
+Furthermore, let $\dfrac a b \in \Q$.
+Then, $a b \in \Q$.
+{{:Euclid:Proposition/X/19}}
+\end{theorem}<|endoftext|>
+\section{Apotome is Irrational}
+Tags: Apotome
+
+\begin{theorem}
+Every [[Definition:Apotome|apotome]] is [[Definition:Irrational Number|irrational]], i.e.:
+: $\displaystyle \forall a, b \in \set {x \in \R_{>0} : x^2 \in \Q}: \paren {\frac a b \notin \Q \land \paren {\frac a b}^2 \in \Q} \implies \paren {\paren {a - b} \notin \Q \land \paren {a - b}^2 \notin \Q}$
+{{:Euclid:Proposition/X/73}}
+\end{theorem}
+
+\begin{proof}
+:[[File:Euclid-X-73.png|300px]]
+Let $AB$ be a [[Definition:Rational Line Segment|rational straight line]].
+Let a [[Definition:Rational Line Segment|rational straight line]] $BC$ which is [[Definition:Commensurable in Square Only|commensurable in square only]] with $AB$ be cut off from $AB$.
+We have that $AB$ is [[Definition:Incommensurable in Length|incommensurable in length]] with $BC$.
+We also have:
+:$AB : BC = AB^2 : AB \cdot AC$
+Therefore from {{EuclidPropLink|book = X|prop = 11|title = Commensurability of Elements of Proportional Magnitudes}}:
+:$AB^2$ is [[Definition:Incommensurable|incommensurable]] with $AB \cdot AC$.
+But from {{EuclidPropLink|book = X|prop = 15|title = Commensurability of Sum of Commensurable Magnitudes}}:
+:$AB^2 + BC^2$ is [[Definition:Commensurable|commensurable]] with $AB^2$.
+and from {{EuclidPropLink|book = X|prop = 6|title = Magnitudes with Rational Ratio are Commensurable}}:
+:$2 \cdot AB \cdot AC$ is [[Definition:Commensurable|commensurable]] with $AB \cdot AC$.
+From {{EuclidPropLink|book = II|prop = 7|title = Square of Difference}}:
+:$AB^2 + BC^2 = 2 \cdot AB \cdot AC + CA^2$
+and so from:
+:{{EuclidPropLink|book = X|prop = 13|title = Commensurable Magnitudes are Incommensurable with Same Magnitude}}
+and:
+:{{EuclidPropLink|book = X|prop = 16|title = Incommensurability of Sum of Incommensurable Magnitudes}}
+it follows that:
+:$AB^2 + BC^2$ is [[Definition:Incommensurable|incommensurable]] with $AC^2$.
+But $AB$ and $BC$ are [[Definition:Rational Line Segment|rational]].
+Therefore $AC$ is [[Definition:Irrational Line Segment|irrational]].
+Such a [[Definition:Line Segment|straight line]] is known as an [[Definition:Apotome|apotome]].
+{{qed}}
+{{Euclid Note|73|X}}
+\end{proof}<|endoftext|>
+\section{Construction of Apotome is Unique}
+Tags: Euclidean Number Theory
+
+\begin{theorem}
+Let $D$ be the domain $\left\{{x \in \R_{>0} : x^2 \in \Q}\right\}$, the [[Definition:Rationally Expressible Number|rationally expressible numbers]].
+Let $a, b \in D$ be two [[Definition:Rationally Expressible Number|rationally expressible numbers]] such that $a - b$ is an [[Definition:Apotome|apotome]].
+Then, there exists only one $x \in D$ such that $a - b + x$ and $a$ are [[Definition:Commensurable in Square Only|commensurable in square only]].
+{{:Euclid:Proposition/X/79}}
+\end{theorem}
+
+\begin{proof}
+:[[File:Euclid-X-79.png|300px]]
+Let $AB$ be an [[Definition:Apotome|apotome]].
+Let $BC$ be added to $AB$ so that $AC$ and $CB$ are [[Definition:Rational Line Segment|rational straight lines]] which are [[Definition:Commensurable in Square Only|commensurable in square only]].
+It is to be proved that no other [[Definition:Rational Line Segment|rational straight line]] can be added to $AB$ which is [[Definition:Commensurable in Square Only|commensurable in square only]] with the whole.
+Suppose $BD$ can be added to $AB$ so as to fulfil the conditions stated.
+Then $AD$ and $DB$ are [[Definition:Rational Line Segment|rational straight lines]] which are [[Definition:Commensurable in Square Only|commensurable in square only]].
+From {{EuclidPropLink|book = II|prop = 7|title = Square of Difference}}:
+:$AD^2 + DB^2 - 2 \cdot AD \cdot DB = AC^2 + CB^2 - 2 \cdot AC \cdot CB = AB^2$
+Therefore:
+:$AD^2 + DB^2 - AC^2 + CB^2 = 2 \cdot AD \cdot DB - 2 \cdot AC \cdot CB$
+But $AD^2 + DB^2$ and $AC^2 + CB^2$ are both [[Definition:Rational Area|rational]].
+Therefore $AD^2 + DB^2 - AC^2 + CB^2$ is [[Definition:Rational Area|rational]].
+Therefore $2 \cdot AD \cdot DB - 2 \cdot AC \cdot CB$ is [[Definition:Rational Area|rational]].
+But from {{EuclidPropLink|book = X|prop = 21|title = Medial is Irrational}}:
+:$2 \cdot AD \cdot DB$ and $2 \cdot AC \cdot CB$ are both [[Definition:Medial Area|medial]].
+From {{EuclidPropLink|book = X|prop = 26|title = Medial Area not greater than Medial Area by Rational Area}}:
+:a [[Definition:Medial Area|medial area]] does not exceed a [[Definition:Medial Area|medial area]] by a [[Definition:Rational Area|rational area]].
+Therefore no other [[Definition:Rational Line Segment|rational straight line]] can be added to $AB$ which is [[Definition:Commensurable in Square Only|commensurable in square only]] with the whole.
+Therefore only one [[Definition:Rational Line Segment|rational straight line]] can be added to $AB$ which is [[Definition:Commensurable in Square Only|commensurable in square only]] with the whole.
+{{qed}}
+{{Euclid Note|79|X}}
+\end{proof}<|endoftext|>
+\section{Lower Bound of Natural Logarithm}
+Tags: Real Analysis, Logarithms, Inequalities, Lower Bound of Natural Logarithm
+
+\begin{theorem}
+:$\forall x \in \R_{>0}: 1 - \dfrac 1 x \le \ln x$
+where $\ln x$ denotes the [[Definition:Natural Logarithm|natural logarithm]] of $x$.
+\end{theorem}
+
+\begin{proof}
+Let $x > 0$.
+{{begin-eqn}}
+{{eqn | l = x - 1 
+      | o = \ge
+      | r = \ln x
+      | c = [[Upper Bound of Natural Logarithm]]
+}}
+{{eqn | ll= \leadsto
+      | l = \frac 1 x -1
+      | o = \ge
+      | r = \ln \frac 1 x
+      | c = putting $\frac 1 x$ into the above inequality
+}}
+{{eqn | r = -\ln x
+      | c = [[Logarithm of Reciprocal]]
+}}
+{{eqn | ll= \leadsto
+      | l = 1 - \frac 1 x
+      | o = \le
+      | r = \ln x
+      | c = multiplying throughout by $-1$
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}
+
+\begin{proof}
+Let $x > 0$.
+Note that:
+:$1 - \dfrac 1 x \le \ln x$
+is [[Definition:Logically Equivalent|logically equivalent]] to:
+:$1 - \dfrac 1 x - \ln x \le 0$
+Let $f \left({x}\right) = 1 - \dfrac 1 x - \ln x$.
+Then:
+{{begin-eqn}}
+{{eqn | l = f \left({x}\right)
+      | r = 1 - \dfrac 1 x - \ln x
+}}
+{{eqn | ll= \implies
+      | l = f' \left({x}\right)
+      | r = \frac 1 {x^2} - \frac 1 x
+      | c = [[Derivative of Constant]], [[Power Rule for Derivatives]], [[Derivative of Natural Logarithm Function]]
+}}
+{{eqn | r = \frac {1 - x} {x^2}
+}}
+{{eqn | ll= \implies
+      | l = f'' \left({x}\right)
+      | r = - \frac 2 {x^3} + \frac 1 {x^2}
+      | c = [[Power Rule for Derivatives]]
+}}
+{{end-eqn}} 
+Note that $f' \left({1}\right) = 0$.
+Also, $f'' \left({1}\right) < 0$.
+So by the [[Second Derivative Test]], $x = 1$ is a [[Definition:Local Maximum|local maximum]].
+On $\left ({0 \,.\,.\, 1} \right)$:
+: $f' \left({x}\right) > 0$
+By [[Derivative of Monotone Function]], $f$ is [[Definition:Strictly Increasing Real Function|strictly increasing]] on that [[Definition:Open Real Interval|interval]].
+On $\left ({1 \,.\,.\, +\infty} \right)$:
+:$f'\left({x}\right) < 0$
+By [[Derivative of Monotone Function]], $f$ is [[Definition:Strictly Decreasing Real Function|strictly decreasing]] on that [[Definition:Unbounded Open Real Interval|interval]].
+So $x = 1$ yields a [[Definition:Global Maximum|global maximum]], at which by [[Logarithm of 1 is 0]]:
+:$f \left({1}\right) = 1 - 1 - 0 = 0$
+That is:
+: $\forall x > 0: f \left({x}\right) \le 0$
+and so by definition of $f \left({x}\right)$:
+:$1 - \dfrac 1 x - \ln x \le 0$
+{{qed}}
+\end{proof}
+
+\begin{proof}
+Let $\sequence {f_n}$ be the [[Definition:Sequence|sequence]] of [[Definition:Mapping|mappings]] $f_n: \R_{>0} \to \R$ defined as:
+:$\map {f_n} x = n \paren {\sqrt [n] x - 1 }$
+Fix $x \in \R_{>0}$.
+We first show that
+$\forall n \in \N : 1 - \dfrac 1 x \le < n \paren {\sqrt [n] x - 1}$
+Let $n \in \N$.
+From [[Sum of Geometric Sequence]]:
+:$\sqrt [n] x - 1 = \dfrac {x - 1} {1 + \sqrt [n] x + \sqrt [n] x^2 + \cdots + \sqrt [n] x^{n - 1} }$
+=== Case 1: $0 < x < 1$ ===
+{{begin-eqn}}
+{{eqn | l = 0 < x < 1
+      | o = \leadsto
+      | r = \forall k < n: \sqrt [n] x^{n - k} > x > 0
+      | c = [[Power Function on Base between Zero and One is Strictly Decreasing/Rational Number]]
+}}
+{{eqn | o = \leadsto
+      | r = 0 < n x < \sum_{k = 0}^{n - 1} \sqrt [n] x^{n - k}
+      | c = [[Real Number Ordering is Compatible with Addition]]
+}}
+{{eqn | o = \leadsto
+      | r = \dfrac 1 {1 + \sqrt [n] x + \sqrt [n] x^2 + \cdots + \sqrt [n] x^{n - 1} } > \dfrac 1 {n x}
+      | c = [[Ordering of Reciprocals]]
+}}
+{{eqn | o = \leadsto
+      | r = \dfrac {x - 1} {n x} < \dfrac {x - 1} {1 + \sqrt [n] x + \sqrt [n] x^2 + \cdots + \sqrt [n] x^{n - 1} }
+      | c = [[Order of Real Numbers is Dual of Order of their Negatives]]
+}}
+{{eqn | o = \leadsto
+      | r = \dfrac {x - 1} {n x} < \sqrt [n] x - 1
+      | c = [[Sum of Geometric Sequence]]
+}}
+{{eqn | o = \leadsto
+      | r = 1 - \dfrac 1  x < n \paren {\sqrt [n] x - 1}
+      | c = [[Real Number Ordering is Compatible with Multiplication]]
+}}
+{{end-eqn}}
+{{qed|lemma}}
+=== Case 2: $x = 1$ ===
+{{begin-eqn}}
+{{eqn | l = \dfrac {x - 1} x
+      | r = 0
+}}
+{{eqn | r = \sqrt [n] 1 - 1
+}}
+{{end-eqn}}
+{{qed|lemma}}
+=== Case 3: $x > 1$ ===
+{{begin-eqn}}
+{{eqn | l = x > 1
+      | o = \leadsto
+      | r = \forall k < n: 1 < \sqrt [n] x^{n - k} < x
+      | c = [[Power Function on Base Greater than One is Strictly Increasing/Rational Number]]
+}}
+{{eqn | o = \leadsto
+      | r = 0 < \sum_{k \mathop = 0}^{n - 1} \sqrt [n] x^{n - k} < n x
+      | c = [[Real Number Ordering is Compatible with Addition]]
+}}
+{{eqn | o = \leadsto
+      | r = 0 < \dfrac 1 {n x} < \dfrac 1 {1 + \sqrt [n] x + \sqrt [n] x^2 + \cdots + \sqrt [n] x^{n - 1} }
+      | c = [[Ordering of Reciprocals]]
+}}
+{{eqn | o = \leadsto
+      | r = \dfrac {x - 1} {n x} < \dfrac {x - 1} {1 + \sqrt [n] x + \sqrt [n] x^2 + \cdots + \sqrt [n] x^{n - 1} }
+      | c = [[Real Number Ordering is Compatible with Multiplication]]
+}}
+{{eqn | o = \leadsto
+      | r = \dfrac {x - 1} {n x} < \sqrt [n] x - 1
+      | c = [[Sum of Geometric Sequence]]
+}}
+{{eqn | o = \leadsto
+      | r = 1 - \dfrac 1 x < n \paren {\sqrt [n] x - 1 }
+      | c = [[Real Number Ordering is Compatible with Multiplication]]
+}}
+{{end-eqn}}
+{{qed|lemma}}
+Thus:
+:$\forall n \in \N : 1 - \dfrac 1 x \le n \paren {\sqrt [n] x - 1 }$
+by [[Proof by Cases]].
+Thus:
+:$\displaystyle 1 - \dfrac 1 x \le \lim_{n \mathop \to \infty} n \paren {\sqrt [n] x - 1 }$
+from [[Limit of Bounded Convergent Sequence is Bounded]].
+Hence the result, from the [[Definition:Real Natural Logarithm|definition of $\ln$]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Bounds of Natural Logarithm}
+Tags: Analysis, Logarithms, Inequalities
+
+\begin{theorem}
+Let $\ln y$ be the [[Definition:Natural Logarithm|natural logarithm]] of $y$ where $y \in \R_{>0}$.
+Then $\ln$ satisfies the [[Definition:Compound Inequality|compound inequality]]:
+:$\displaystyle 1 - \frac 1 y \le \ln y \le y - 1$
+\end{theorem}
+
+\begin{proof}
+* [[Upper Bound of Natural Logarithm]]: $\ln y \le y - 1$
+* [[Lower Bound of Natural Logarithm]]: $1 - \dfrac 1 y \le \ln y$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Area contained by Apotome and Binomial Straight Line Commensurable with Terms of Apotome and in same Ratio/Porism}
+Tags: Euclidean Number Theory
+
+\begin{theorem}
+{{:Euclid:Proposition/X/114/Porism}}
+\end{theorem}
+
+\begin{proof}
+Directly apparent from the construction.
+{{qed}}
+{{Euclid Note|114|X}}
+\end{proof}<|endoftext|>
+\section{Construction of Solid Angle from Three Plane Angles any Two of which are Greater than Other Angle/Lemma}
+Tags: Solid Geometry
+
+\begin{theorem}
+{{:Euclid:Proposition/XI/23/Lemma}}
+\end{theorem}
+
+\begin{proof}
+:[[File:Euclid-XI-23b.png|400px]]
+Let the [[Definition:Straight Line|straight lines]] $AB$ and $LO$ be set out.
+Le $AB > LO$.
+Let the [[Definition:Semicircle|semicircle]] $ACB$ be described on $AB$.
+:[[File:Euclid-XI-23-Lemma.png|400px]]
+Using {{EuclidPropLink|book = IV|prop = 1|title = Fitting Chord Into Circle}}:
+:Let $AC = LO$ be fitted into the [[Definition:Semicircle|semicircle]] $ACB$.
+Let $BC$ be joined.
+From {{EuclidPropLink|book = III|prop = 31|title = Relative Sizes of Angles in Segments}}:
+:$\angle ACB$ is a [[Definition:Right Angle|right angle]].
+Therefore from {{EuclidPropLink|book = I|prop = 47|title = Pythagoras's Theorem}}:
+:$AB^2 = AC^2 + CB^2$
+Hence:
+:$AB^2 - AC^2 = CB^2$
+But $AC = LO$.
+Therefore:
+:$AB^2 - LO^2 = CB^2$
+So if we then cut of $OR = BC$:
+:$AB^2 - LO^2 = OR^2$
+{{qed}}
+{{Euclid Note|23|XI}}
+\end{proof}<|endoftext|>
+\section{Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides/Porism}
+Tags: Parallelepipeds
+
+\begin{theorem}
+{{:Euclid:Proposition/XI/33/Porism}}
+\end{theorem}
+
+\begin{proof}
+Apparent from the construction.
+{{qed}}
+{{Euclid Note|33|XI}}
+\end{proof}<|endoftext|>
+\section{Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles/Porism}
+Tags: Solid Geometry
+
+\begin{theorem}
+{{:Euclid:Proposition/XI/35/Porism}}
+\end{theorem}
+
+\begin{proof}
+Directly apparent from the construction.
+{{qed}}
+{{Euclid Note|35|XI}}
+\end{proof}<|endoftext|>
+\section{Areas of Circles are as Squares on Diameters/Lemma}
+Tags: Circles
+
+\begin{theorem}
+{{:Euclid:Proposition/XII/2/Lemma}}
+\end{theorem}
+
+\begin{proof}
+:[[File:Euclid-XII-2.png|800px]]
+Let it be contrived that:
+:$S : ABCD = EFGH : T$
+for some [[Definition:Area|area]] $T$.
+It is to be demonstrated that $T$ is less than the [[Definition:Circle|circle]] $ABCD$.
+From the relationship:
+:$S : ABCD = EFGH : T$
+it follows from {{EuclidPropLink|book = V|prop = 16|title = Proportional Magnitudes are Proportional Alternately}} that:
+:$S : EFGH = ABCD : T$
+But:
+:$S > EFGH$
+Therefore:
+:$ABCD > T$
+Hence the result.
+{{qed}}
+{{Euclid Note|2|XII}}
+\end{proof}<|endoftext|>
+\section{Tetrahedra are Equal iff Bases are Reciprocally Proportional to Heights}
+Tags: Tetrahedra
+
+\begin{theorem}
+{{:Euclid:Proposition/XII/9}}
+\end{theorem}
+
+\begin{proof}
+:[[File:Euclid-XII-9.png|500px]]
+Let $ABCG$ be a [[Definition:Tetrahedron|tetrahedron]] whose [[Definition:Base of Pyramid|base]] is $\triangle ABC$ and whose [[Definition:Apex of Pyramid|apex]] is $G$.
+Let $DEFH$ be a [[Definition:Tetrahedron|tetrahedron]] whose [[Definition:Base of Pyramid|base]] is $\triangle DEF$ and whose [[Definition:Apex of Pyramid|apex]] is $H$.
+Let $ABCG$ and $DEFH$ be equal in [[Definition:Volume|volume]].
+It is to be demonstrated that the [[Definition:Area|areas]] of their [[Definition:Base of Pyramid|bases]] is [[Definition:Reciprocal Proportion|reciprocally proportional]] to their [[Definition:Height of Pyramid|heights]].
+Let the [[Definition:Parallelepiped|parallelepipeds]] $BGML$ and $EHQP$ be completed.
+We have that $ABCG$ is equal to $DEFH$.
+From:
+:{{EuclidPropLink|book = XI|prop = 28|title = Parallelepiped cut by Plane through Diagonals of Opposite Planes is Bisected}}
+and
+:{{EuclidPropLink|book = XII|prop = 7|title = Prism on Triangular Base divided into Three Equal Tetrahedra}}
+it follows that:
+:$BGML = 6 \cdot ABCG$
+and:
+:$EHQP = 6 \cdot DEFH$
+Therefore:
+:$BGML = EHQP$
+But from {{EuclidPropLink|book = XI|prop = 34|title = Parallelepipeds are of Equal Volume iff Bases are in Reciprocal Proportion to Heights}}:
+:$BM : EQ = h \left({EHQP}\right) : h \left({BGML}\right)$
+where:
+:$BM$ and $EQ$ are the [[Definition:Base of Parallelepiped|bases]] of $BGML$ and $EHQP$ respectively
+:$h \left({EHQP}\right)$ denotes the [[Definition:Height of Parallelepiped|height]] of $EHQP$
+:$h \left({BGML}\right)$ denotes the [[Definition:Height of Parallelepiped|height]] of $BGML$.
+But from {{EuclidPropLink|book = I|prop = 34|title = Opposite Sides and Angles of Parallelogram are Equal}}:
+:$BM : EQ = \triangle ABC : \triangle DEF$
+Therefore by {{EuclidPropLink|book = V|prop = 11|title = Equality of Ratios is Transitive}}:
+:$\triangle ABC : \triangle DEF = h \left({EHQP}\right) : h \left({BGML}\right)$
+But:
+:$h \left({EHQP}\right) = h \left({DEFH}\right)$
+:$h \left({BGML}\right) = h \left({ABCG}\right)$
+where:
+:$h \left({ABCG}\right)$ denotes the [[Definition:Height of Pyramid|height]] of $ABCG$
+:$h \left({DEFH}\right)$ denotes the [[Definition:Height of Pyramid|height]] of $DEFH$.
+Therefore:
+:$\triangle ABC : \triangle DEF = h \left({DEFH}\right) : h \left({ABCG}\right)$
+That is the [[Definition:Base of Parallelepiped|bases]] of $ABCG$ and $DEFH$ are [[Definition:Reciprocal Proportion|reciprocally proportional]] to their [[Definition:Height of Pyramid|heights]].
+{{qed|lemma}}
+Let $ABCG$ and $DEFH$ be [[Definition:Tetrahedron|tetrahedra]] whose [[Definition:Base of Parallelepiped|bases]] are [[Definition:Reciprocal Proportion|reciprocally proportional]] to their [[Definition:Height of Pyramid|heights]]:
+:$\triangle ABC : \triangle DEF = h \left({DEFH}\right) : h \left({ABCG}\right)$
+It is to be shown that $ABCG$ is equal to $DEFH$.
+Let the [[Definition:Parallelepiped|parallelepipeds]] $BGML$ and $EHQP$ be completed.
+We have that:
+:$\triangle ABC : \triangle DEF = h \left({DEFH}\right) : h \left({ABCG}\right)$
+while:
+:$\triangle ABC : \triangle DEF = BM : EQ$
+where $BM$ and $EQ$ are the [[Definition:Base of Parallelepiped|bases]] of $BGML$ and $EHQP$ respectively.
+Therefore by {{EuclidPropLink|book = V|prop = 11|title = Equality of Ratios is Transitive}}:
+:$BM : EQ = h \left({EHQP}\right) : h \left({BGML}\right)$
+From {{EuclidPropLink|book = XI|prop = 34|title = Parallelepipeds are of Equal Volume iff Bases are in Reciprocal Proportion to Heights}}:
+:$BGML = EHQP$
+But from:
+:{{EuclidPropLink|book = XI|prop = 28|title = Parallelepiped cut by Plane through Diagonals of Opposite Planes is Bisected}}
+and
+:{{EuclidPropLink|book = XII|prop = 7|title = Prism on Triangular Base divided into Three Equal Tetrahedra}}
+it follows that:
+:$BGML = 6 \cdot ABCG$
+and:
+:$EHQP = 6 \cdot DEFH$
+Therefore:
+:$ABCG = DEFH$
+{{qed}}
+{{Euclid Note|9|XII}}
+\end{proof}<|endoftext|>
+\section{Construction of Polyhedron in Outer of Concentric Spheres/Porism}
+Tags: Spheres, Polyhedra
+
+\begin{theorem}
+{{:Euclid:Proposition/XII/17/Porism}}
+\end{theorem}
+
+\begin{proof}
+:[[File:Euclid-XII-17.png|800px]]
+We have that the [[Definition:Polyhedron|polyhedron]] as constructed in {{EuclidPropLink|book = XII|prop = 17|title = Construction of Polyhedron in Outer of Concentric Spheres}} is divided into [[Definition:Pyramid|pyramids]] which are [[Definition:Similar Solid Figures|similar]] in number and arrangement.
+But from {{EuclidPropLink|book = XII|prop = 8|qual=Porism|title = Volumes of Similar Tetrahedra are in Triplicate Ratio of Corresponding Sides}}:
+:[[Definition:Similar Solid Figures|similar]] [[Definition:Pyramid|pyramids]] are to one another in the [[Definition:Triplicate Ratio|triplicate ratio]] of their corresponding [[Definition:Edge of Polyhedron|sides]].
+Therefore the [[Definition:Pyramid|pyramid]] whose [[Definition:Base of Pyramid|base]] is $KBPS$ and whose [[Definition:Apex of Pyramid|apex]] is $A$ has to the similarly arranged [[Definition:Pyramid|pyramid]] in the other [[Definition:Sphere (Geometry)|sphere]] the [[Definition:Triplicate Ratio|triplicate ratio]] of their corresponding [[Definition:Edge of Polyhedron|sides]]: that is, $AB$ to the [[Definition:Radius of Sphere|radius]] of the other [[Definition:Sphere (Geometry)|sphere]].
+The same applies to all the other [[Definition:Pyramid|pyramids]].
+It follows from {{EuclidPropLink|book = V|prop = 12|title = Sum of Components of Equal Ratios}}:
+:the sum total of all the [[Definition:Pyramid|pyramids]] that form the whole [[Definition:Polyhedron|polyhedron]] in the one [[Definition:Sphere (Geometry)|sphere]] has to the sum total of all the [[Definition:Pyramid|pyramids]] that form the whole [[Definition:Polyhedron|polyhedron]] in the other [[Definition:Sphere (Geometry)|sphere]] the [[Definition:Triplicate Ratio|triplicate ratio]] of the [[Definition:Radius of Sphere|radii]] of the [[Definition:Sphere (Geometry)|spheres]].
+That is, the [[Definition:Triplicate Ratio|triplicate ratio]] of the [[Definition:Diameter of Sphere|diameters]] of the [[Definition:Sphere (Geometry)|spheres]].
+{{qed}}
+{{Euclid Note|17|XII}}
+\end{proof}<|endoftext|>
+\section{Definition:Elevated/Point}
+Tags: Definitions: Solid Geometry
+
+\begin{theorem}
+Let $P$ be a [[Definition:Plane|plane]].
+An '''elevated point (relative to $P$)''' is a [[Definition:Point|point]] in [[Definition:Ordinary Space|space]] which does not lie on $P$.
+\end{theorem}<|endoftext|>
+\section{Definition:Elevated/Line}
+Tags: Definitions: Solid Geometry
+
+\begin{theorem}
+Let $P$ be a [[Definition:Plane|plane]].
+An '''elevated line (relative to $P$)''' is a [[Definition:Line|line]] in [[Definition:Ordinary Space|space]] which does not lie entirely in $P$.
+\end{theorem}<|endoftext|>
+\section{Mordell's Theorem}
+Tags: Algebraic Geometry
+
+\begin{theorem}
+Let $C$ be an [[Definition:Elliptic Curve|elliptic curve]] over $\Q$.
+Then the [[Definition:Group of Rational Points|group of rational points]] on $C$ is a [[Definition:Finitely Generated Abelian Group|finitely generated abelian group]].
+\end{theorem}
+
+\begin{proof}
+{{ProofWanted}}
+{{Namedfor|Louis Joel Mordell|cat = Mordell}}
+\end{proof}<|endoftext|>
+\section{Common Ratio in Rational Geometric Sequence is Rational}
+Tags: Geometric Sequences
+
+\begin{theorem}
+Let $\sequence {a_k}$ be a [[Definition:Geometric Sequence|geometric sequence]] whose [[Definition:Term of Geometric Sequence|terms]] are [[Definition:Rational Number|rational]].
+Then the [[Definition:Common Ratio|common ratio]] of $\sequence {a_k}$ is [[Definition:Rational Number|rational]].
+\end{theorem}
+
+\begin{proof}
+Let $r$ be the [[Definition:Common Ratio|common ratio]] of $\sequence {a_k}$.
+Let $p, q$ be consecutive [[Definition:Term of Geometric Sequence|terms]] of $r$.
+[[Definition:By Hypothesis|By hypothesis]] $p, q \in \Q$.
+Then, by definition of [[Definition:Geometric Sequence|geometric sequence]]:
+:$q = r p$
+It follows that:
+:$r = \dfrac q p$
+From [[Rational Numbers form Field]], $\Q$ is [[Definition:Closed Algebraic Structure|closed]] under [[Definition:Division|division]].
+Thus $r \in \Q$ and hence the result.
+{{qed}}
+[[Category:Geometric Sequences]]
+qoxjqks99ap78b23fvywcsbpiwnffa0
+\end{proof}<|endoftext|>
+\section{Common Ratio in Integer Geometric Sequence is Rational}
+Tags: Geometric Sequences of Integers
+
+\begin{theorem}
+Let $\sequence {a_k}$ be a [[Definition:Geometric Sequence|geometric sequence]] whose [[Definition:Term of Geometric Sequence|terms]] are all [[Definition:Integer|integers]].
+Then the [[Definition:Common Ratio|common ratio]] of $\sequence {a_k}$ is [[Definition:Rational Number|rational]].
+\end{theorem}
+
+\begin{proof}
+From [[Integers form Subdomain of Rationals]] it follows that $a_k \in \Q$ for all $0 \le k \le n$.
+The result follows from [[Common Ratio in Rational Geometric Sequence is Rational]].
+{{qed}}
+[[Category:Geometric Sequences of Integers]]
+0qft406itmha5ood9xvu2bzzz0xivkq
+\end{proof}<|endoftext|>
+\section{Form of Geometric Sequence of Integers}
+Tags: Geometric Sequences of Integers
+
+\begin{theorem}
+Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a [[Definition:Geometric Sequence|geometric sequence]] of [[Definition:Length of Sequence|length]] $n + 1$ consisting of [[Definition:Integer|integers]] only.
+Then the $j$th [[Definition:Term of Geometric Sequence|term]] of $P$ is given by:
+:$a_j = k p^{n - j} q^j$
+where:
+: the [[Definition:Common Ratio|common ratio]] of $P$ expressed in [[Definition:Canonical Form of Rational Number|canonical form]] is $\dfrac q p$
+: $k$ is an [[Definition:Integer|integer]].
+\end{theorem}
+
+\begin{proof}
+Let $r$ be the [[Definition:Common Ratio of Geometric Sequence|common ratio]] of $P$.
+From [[Common Ratio in Integer Geometric Sequence is Rational]], $r$ is a [[Definition:Rational Number|rational number]].
+Let $r = \dfrac q p$ be in [[Definition:Canonical Form of Rational Number|canonical form]].
+Thus, by definition:
+:$p \perp q$
+Let $a$ be the first [[Definition:Term of Geometric Sequence|term]] of $P$.
+Then the [[Definition:Sequence|sequence]] $P$ is:
+:$P = \paren {a, a \dfrac q p, a \dfrac {q^2} {p^2}, \ldots, a \dfrac {q^n} {p^n} }$
+All the elements of $P$ are [[Definition:Integer|integers]], so, in particular:
+:$a \dfrac {q^n} {p^n} \in \Z$
+We have that: 
+:$p \perp q$
+From [[Powers of Coprime Numbers are Coprime]]:
+:$q^n \perp p^n$
+and so from [[Euclid's Lemma]]:
+:$p^n \divides a$
+Thus:
+:$a = k p^n$
+for some $k \in \Z$, and so:
+:$P = \paren {k p^n, k q p^{n - 1}, k q^2 p^{n - 2}, \ldots, k q^{n - 1} p, k q^n}$
+{{qed}}
+[[Category:Geometric Sequences of Integers]]
+bhfwiu2z97dxzp20i8vusbgjraa7agi
+\end{proof}<|endoftext|>
+\section{Geometric Sequence of Integers with Integer Common Ratio}
+Tags: Geometric Sequences of Integers
+
+\begin{theorem}
+Let $P = \sequence {a_j}_{1 \mathop \le j \mathop \le n}$ be a [[Definition:Geometric Sequence|geometric sequence]] of [[Definition:Length of Sequence|length]] $n$ consisting entirely of [[Definition:Integer|integers]].
+Let $r$ be the [[Definition:Common Ratio of Geometric Sequence|common ratio]] of $P$.
+Then $r$ is an [[Definition:Integer|integer]] {{iff}}:
+:$\forall i, j \in \set {1, 2, \ldots, n}, i \le j: a_i \divides a_j$
+That is, [[Definition:Term of Geometric Sequence|terms]] of $P$ [[Definition:Divisor of Integer|divide]] later [[Definition:Term of Geometric Sequence|terms]] of $P$ {{iff}} $r$ is an [[Definition:Integer|integer]].
+\end{theorem}
+
+\begin{proof}
+=== Necessary Condition ===
+Let $r$ be an [[Definition:Integer|integer]].
+By definition of [[Definition:Geometric Sequence|geometric sequence]], the [[Definition:Term of Geometric Sequence|terms]] of $P$ are in the form:
+:$a_k = b r^{k - 1}$
+where $b$ and $k$ are [[Definition:Integer|integers]].
+It follows from [[Integer Multiplication is Closed]] that $a_k$ is an [[Definition:Integer|integer]].
+{{qed|lemma}}
+=== Sufficient Condition ===
+From [[Common Ratio in Integer Geometric Sequence is Rational]], $r$ is a [[Definition:Rational Number|rational number]].
+Let $r$ be expressed in [[Definition:Canonical Form of Rational Number|canonical form]] as:
+:$r = \dfrac p q$
+where, by definition of [[Definition:Canonical Form of Rational Number|canonical form]], $p \perp q$, that is, $p$ is [[Definition:Coprime Integers|coprime]] to $q$.
+From [[Form of Geometric Sequence of Integers]], the [[Definition:Term of Geometric Sequence|terms]] of $P$ are in the form:
+:$(1): \quad a_j = k q^{j - 1} p^{n - j}$
+Let $a_i \divides a_j$.
+From [[Powers of Coprime Numbers are Coprime]]:
+:$(2): \quad q^{j - i} \perp p^{j - i}$
+Then:
+{{begin-eqn}}
+{{eqn | l = a_i
+      | o = \divides
+      | r = a_j
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = k q^{i - 1} p^{n - i}
+      | o = \divides
+      | r = k q^{j - 1} p^{n - j}
+      | c = from $(1)$
+}}
+{{eqn | ll= \leadsto
+      | lo= \exists m \in \Z:
+      | l = m \paren {k q^{j - 1} p^{n - i} } q^{j - i}
+      | r = \paren {k q^{j - 1} p^{n - i} } p^{j - i}
+      | c = {{Defof|Divisor of Integer}}
+}}
+{{eqn | ll= \leadsto
+      | l = m q^{j - i}
+      | r = p^{j - i}
+      | c = dividing both sides by $k q^{j - 1} p^{n - i}$
+}}
+{{eqn | ll= \leadsto
+      | l = q^{j - i}
+      | o = \divides
+      | r = p^{j - i}
+      | c = {{Defof|Divisor of Integer}}
+}}
+{{end-eqn}}
+But from $(2)$:
+:$q^{j - i} \perp p^{j - i}$
+Thus $q^{j - i} \divides p^{j - i}$ can happen only when $q^{j - 1} = q = 1$.
+That is, when $r$ is an [[Definition:Integer|integer]].
+{{qed}}
+[[Category:Geometric Sequences of Integers]]
+3k4xydo9q8ax698lep4fppvf96to2eu
+\end{proof}<|endoftext|>
+\section{Integers are Coprime iff Powers are Coprime}
+Tags: Coprime Integers
+
+\begin{theorem}
+Let $a, b \in \Z$ be [[Definition:Integer|integers]].
+Then:
+:$a \perp b \iff \forall n \in \N: a^n \perp b^n$
+That is, two [[Definition:Integer|integers]] are [[Definition:Coprime Integers|coprime]] {{iff}} all their [[Definition:Integer Power|positive integer powers]] are [[Definition:Coprime Integers|coprime]].
+\end{theorem}
+
+\begin{proof}
+The forward implication is shown in [[Powers of Coprime Numbers are Coprime]].
+The reverse implication is shown by substituting $n = 1$.
+{{qed}}
+[[Category:Coprime Integers]]
+8k2ky39jzctjiltp3egexjushvinn3s
+\end{proof}<|endoftext|>
+\section{Successor Mapping of Peano Structure has no Fixed Point}
+Tags: Peano's Axioms
+
+\begin{theorem}
+Let $\PP = \struct {P, s, 0}$ be a [[Definition:Peano Structure|Peano structure]].
+Then:
+:$\forall n \in P: \map s n \ne n$
+That is, the [[Definition:Successor Mapping|successor mapping]] has no [[Definition:Fixed Point|fixed points]].
+\end{theorem}
+
+\begin{proof}
+Let $T$ be the set:
+:$T = \set {n \in P: \map s n \ne n}$
+We will use [[Axiom:Peano's Axioms|Axiom $(\text P 5)$]] to prove that $T = P$.
+=== Part $1$: $0 \in T$ ===
+From [[Axiom:Peano's Axioms|Axiom $(\text P 4)$]], [[Definition:A Fortiori|a fortiori]]:
+:$\map s 0 \ne 0$
+Hence $0 \in T$.
+{{qed|lemma}}
+=== Part $2$: $n \in T \implies \map s n \in T$ ===
+Let $n \in T$.
+Let $m = \map s n$.
+Then $n \ne m$, as $\map s n \ne n$.
+{{AimForCont}} $m \notin T$.
+That is:
+:$\map s m = m$
+Then that would mean:
+:$\map s n = \map s m = m$
+But from [[Axiom:Peano's Axioms|Axiom $(\text P 3)$]], $s$ is [[Definition:Injection|injective]].
+From this [[Definition:Contradiction|contradiction]], it follows that $m \in T$.
+Hence we have that:
+:$n \in T \implies \map s n \in T$
+{{qed|lemma}}
+=== Conclusion ===
+By [[Axiom:Peano's Axioms|Axiom $(\text P 5)$]], we conclude that $T = P$.
+From the definition of $T$, the result follows.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Integer Addition is Cancellable}
+Tags: Integer Addition
+
+\begin{theorem}
+The operation of [[Definition:Integer Addition|addition]] on the [[Definition:Set|set]] of [[Definition:Integer|integers]] $\Z$ is [[Definition:Cancellable|cancellable]]:
+:$\forall x, y, z \in \Z: x + z = y + z \implies x = y$
+\end{theorem}
+
+\begin{proof}
+Let $x = \eqclass {a, b} {}$, $y = \eqclass {c, d} {}$ and $z = \eqclass {e, f} {}$ for some $x, y, z\in \Z$.
+Then:
+{{begin-eqn}}
+{{eqn | l = x + z
+      | r = y + z
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = \eqclass {a, b} {} + \eqclass {e, f} {}
+      | r = \eqclass {c, d} {} + \eqclass {e, f} {}
+      | c =
+}}
+{{eqn | ll= \leadsto
+      | l = \eqclass {a + e, b + f} {}
+      | r = \eqclass {c + e, d + f} {}
+      | c = {{Defof|Integer Addition}}
+}}
+{{eqn | ll= \leadsto
+      | l = a + e
+      | r = c + e
+      | c = {{Defof|Integer}}
+}}
+{{eqn | lo= \land
+      | l = b + f
+      | r = d + f
+      | c = {{Defof|Integer}}
+}}
+{{eqn | ll= \leadsto
+      | l = a
+      | r = c
+      | c = [[Natural Number Addition is Cancellable]]
+}}
+{{eqn | lo= \land
+      | l = b
+      | r = d
+      | c = [[Natural Number Addition is Cancellable]]
+}}
+{{eqn | ll= \leadsto
+      | l = \eqclass {a, b} {}
+      | r = \eqclass {c, d} {}
+      | c = {{Defof|Integer}}
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Open Real Interval is Subset of Closed Real Interval}
+Tags: Real Intervals
+
+\begin{theorem}
+Let $a, b \in \R$ be [[Definition:Real Number|real numbers]].
+Then:
+:$\openint a b \subseteq \closedint a b$
+where:
+:$\openint a b$ is the [[Definition:Open Real Interval|open interval]] between $a$ and $b$
+:$\closedint a b$ is the [[Definition:Closed Real Interval|closed interval]] between $a$ and $b$.
+\end{theorem}
+
+\begin{proof}
+Let $x \in \openint a b$.
+Then by definition of [[Definition:Open Real Interval|open interval]]:
+:$a < x < b$
+Thus:
+:$a \le x \le b$
+and so by definition of [[Definition:Closed Real Interval|closed interval]]:
+:$x \in \closedint a b$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Complement Relative to Subset is Subset of Complement Relative to Superset}
+Tags: Subsets, Relative Complement
+
+\begin{theorem}
+Let $A, B, C$ be [[Definition:Set|sets]] such that $A \subseteq B \subseteq C$.
+Then:
+:$\relcomp B A \subseteq \relcomp C A$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = x
+      | o = \in
+      | r = \relcomp B A
+      | c = 
+}}
+{{eqn | lll=\leadsto
+      | l = x
+      | o = \in
+      | r = B
+      | c = {{Defof|Relative Complement}}
+}}
+{{eqn | ll= \land
+      | l = x
+      | o = \notin
+      | r = A
+      | c = {{Defof|Relative Complement}}
+}}
+{{eqn | lll=\leadsto
+      | l = x
+      | o = \in
+      | r = C
+      | c = {{Defof|Subset}}: $B \subseteq C$
+}}
+{{eqn | ll= \land
+      | l = x
+      | o = \notin
+      | r = A
+      | c = 
+}}
+{{eqn | lll=\leadsto
+      | l = x
+      | o = \in
+      | r = \relcomp C A
+      | c = {{Defof|Relative Complement}}
+}}
+{{eqn | lll=\leadsto
+      | l = \relcomp B A
+      | o = \subseteq
+      | r = \relcomp C A
+      | c = {{Defof|Subset}}
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Complement of Relative Complement is Union with Complement}
+Tags: Subsets, Relative Complement
+
+\begin{theorem}
+Let $A, B, C$ be [[Definition:Set|sets]] such that $A \subseteq B \subseteq C$.
+Then:
+:$\relcomp C {\relcomp B A} = A \cup \relcomp C B$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = \relcomp C {\relcomp B A}
+      | r = C \setminus \paren {B \setminus A}
+      | c = {{Defof|Relative Complement}}
+}}
+{{eqn | r = \paren {C \setminus B} \cup \paren {C \cap A}
+      | c = [[Set Difference with Set Difference is Union of Set Difference with Intersection]]
+}}
+{{eqn | r = \paren {C \setminus B} \cup A
+      | c = [[Intersection with Subset is Subset]]
+}}
+{{eqn | r = A \cup \paren {C \setminus B}
+      | c = [[Union is Commutative]]
+}}
+{{eqn | r = A \cup \relcomp C B
+      | c = {{Defof|Relative Complement}}
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Set Union Preserves Subsets/Families of Sets}
+Tags: Set Union Preserves Subsets, Indexed Families
+
+\begin{theorem}
+Let $I$ be an [[Definition:Indexing Set|indexing set]].
+Let $\family {A_\alpha}_{\alpha \mathop \in I}$ and $\family {B_\alpha}_{\alpha \mathop \in I}$ be [[Definition:Indexed Family of Subsets|indexed families of subsets]] of a [[Definition:Set|set]] $S$.
+Let:
+:$\forall \beta \in I: A_\beta \subseteq B_\beta$
+Then:
+:$\displaystyle \bigcup_{\alpha \mathop \in I} A_\alpha \subseteq \bigcup_{\alpha \mathop \in I} B_\alpha$
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = x
+      | o = \in
+      | r = \bigcup_{\alpha \mathop \in I} A_\alpha
+      | c = 
+}}
+{{eqn | lll=\leadsto
+      | ll= \exists \alpha \in I:
+      | l = x
+      | o = \in
+      | r = A_\alpha
+      | c = {{Defof|Union of Family}}
+}}
+{{eqn | lll=\leadsto
+      | ll= \exists \alpha \in I:
+      | l = x
+      | o = \in
+      | r = B_\alpha
+      | c = {{Defof|Subset}}
+}}
+{{eqn | lll=\leadsto
+      | l = x
+      | o = \in
+      | r = \bigcup_{\alpha \mathop \in I} B_\alpha
+      | c = {{Defof|Union of Family}}
+}}
+{{end-eqn}}
+By definition of [[Definition:Subset|subset]]:
+:$\displaystyle \bigcup_{\alpha \mathop \in I} A_\alpha \subseteq \bigcup_{\alpha \mathop \in I} B_\alpha$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Set Union Preserves Subsets/General Result}
+Tags: Set Union Preserves Subsets
+
+\begin{theorem}
+Let $\mathbb S, \mathbb T$ be [[Definition:Set of Sets|sets of sets]].
+Suppose that for each $S \in \mathbb S$ there exists a $T \in \mathbb T$ such that $S \subseteq T$.
+Then $\bigcup \mathbb S \subseteq \bigcup \mathbb T$.
+\end{theorem}
+
+\begin{proof}
+Let $x \in \bigcup \mathbb S$.
+By the definition of [[Definition:Set Union|union]], there exists an $S \in \mathbb S$ such that $x \in S$.
+By the premise, there exists a $T \in \mathbb T$ such that $S \subseteq T$.
+By the definition of [[Definition:Subset|Subset]]:
+:$x \in T$
+Thus by the definition of [[Definition:Set Union|union]]:
+:$x \in \bigcup \mathbb T$
+{{qed}}
+[[Category:Set Union Preserves Subsets]]
+p76bex02v5vodwdiwzu81ezytmoz9dx
+\end{proof}<|endoftext|>
+\section{Intersection of Family is Subset of Intersection of Subset of Family}
+Tags: Subsets, Set Intersection, Indexed Families
+
+\begin{theorem}
+Let $I$ be an [[Definition:Indexing Set|indexing set]].
+Let $\family {A_\alpha}_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family of Subsets|indexed family of subsets]] of a [[Definition:Set|set]] $S$.
+Let $J \subseteq I$.
+Then:
+:$\displaystyle \bigcap_{\alpha \mathop \in I} A_\alpha \subseteq \bigcap_{\alpha \mathop \in J} A_\alpha$
+where $\displaystyle \bigcap_{\alpha \mathop \in I} A_\alpha$ denotes the [[Definition:Intersection of Family|intersection]] of $\family {A_\alpha}_{\alpha \mathop \in I}$.
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = x
+      | o = \in
+      | r = \bigcap_{\alpha \mathop \in I} A_\alpha
+      | c =
+}}
+{{eqn | lll=\leadsto
+      | ll= \forall \alpha \in I:
+      | l = x
+      | o = \in
+      | r = A_\alpha
+      | c = [[Intersection is Subset/Family of Sets|Intersection is Subset]]
+}}
+{{eqn | lll=\leadsto
+      | ll= \forall \alpha \in J:
+      | l = x
+      | o = \in
+      | r = A_\alpha
+      | c = {{Defof|Subset}}: $J \subseteq I$
+}}
+{{eqn | lll=\leadsto
+      | l = x
+      | o = \in
+      | r = \bigcap_{\alpha \mathop \in J} A_\alpha
+      | c = {{Defof|Intersection of Family}}
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Union of Subset of Family is Subset of Union of Family}
+Tags: Subsets, Set Union, Indexed Families
+
+\begin{theorem}
+Let $I$ be an [[Definition:Indexing Set|indexing set]].
+Let $\family {A_\alpha}_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family of Subsets|indexed family of subsets]] of a [[Definition:Set|set]] $S$.
+Let $J \subseteq I$
+Then:
+:$\displaystyle \bigcup_{\alpha \mathop \in J} A_\alpha \subseteq \bigcup_{\alpha \mathop \in I} A_\alpha$
+where $\displaystyle \bigcup_{\alpha \mathop \in I} A_\alpha$ denotes the [[Definition:Union of Family|union]] of $\family {A_\alpha}_{\alpha \mathop \in I}$.
+\end{theorem}
+
+\begin{proof}
+{{begin-eqn}}
+{{eqn | l = x
+      | o = \in
+      | r = \bigcup_{\alpha \mathop \in J} A_\alpha
+      | c =
+}}
+{{eqn | lll=\leadsto
+      | ll= \exists \alpha \in J:
+      | l = x
+      | o = \in
+      | r = A_\alpha
+      | c = {{Defof|Union of Family}}
+}}
+{{eqn | lll=\leadsto
+      | ll= \exists \alpha \in I:
+      | l = x
+      | o = \in
+      | r = A_\alpha
+      | c = {{Defof|Subset}}: $J \subseteq I$
+}}
+{{eqn | lll=\leadsto
+      | l = x
+      | o = \in
+      | r = \bigcup_{\alpha \mathop \in I} A_\alpha
+      | c = [[Set is Subset of Union/Family of Sets|Set is Subset of Union]]
+}}
+{{end-eqn}}
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Intersection of Open Intervals of Positive Reals is Empty}
+Tags: Real Intervals
+
+\begin{theorem}
+Let $\R_{> 0}$ be the [[Definition:Set|set]] of [[Definition:Strictly Positive Real Number|strictly positive real numbers]].
+For all $x \in \R_{> 0}$, let $A_x$ be the [[Definition:Open Real Interval|open real interval]] $\openint 0 x$.
+Then:
+:$\displaystyle \bigcap_{x \mathop \in \R_{> 0} } A_x = \O$
+\end{theorem}
+
+\begin{proof}
+Let $\displaystyle A = \bigcap_{x \mathop \in \R_{> 0} } A_x$.
+{{AimForCont}} $A \ne \O$.
+Then:
+:$\exists y \in \R_{> 0}: y \in A$
+By definition of [[Definition:Open Real Interval|open interval]]:
+:$y \notin \openint 0 y = A_y$
+and so by definition of [[Definition:Intersection of Family|intersection of family]]:
+:$y \notin A$
+From this [[Proof by Contradiction|contradiction]] it follows that $A$ has no [[Definition:Element|elements]].
+That is:
+:$\displaystyle \bigcap_{x \mathop \in \R_{> 0} } A_x = \O$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Union of Open Intervals of Positive Reals is Set of Strictly Positive Reals}
+Tags: Real Intervals
+
+\begin{theorem}
+Let $\R_{> 0}$ be the [[Definition:Set|set]] of [[Definition:Strictly Positive Real Number|strictly positive real numbers]].
+For all $x \in \R_{> 0}$, let $A_x$ be the [[Definition:Open Real Interval|open real interval]] $\openint 0 x$.
+Then:
+:$\displaystyle \bigcup_{x \mathop \in \R_{> 0} } A_x = \R_{> 0}$
+\end{theorem}
+
+\begin{proof}
+Let $\displaystyle A = \bigcap_{x \mathop \in \R_{> 0} } A_x$.
+Let $y \in A$.
+Then by definition of [[Definition:Union of Family|union of family]]:
+:$\exists x \in \R_{> 0}: y \in A_x$
+As $A_x \subseteq \R_{> 0}$ it follows by definition of [[Definition:Subset|subset]] that:
+:$y \in \R_{> 0}$
+So:
+:$\displaystyle \bigcap_{x \mathop \in \R_{> 0} } A_x \subseteq \R_{> 0}$
+{{qed|lemma}}
+Let $y \in \R_{> 0}$.
+By the [[Archimedean Property]]: 
+:$\exists z \in \N: z > y$
+and so:
+:$y \in A_z$
+That is by definition of [[Definition:Union of Family|union of family]]:
+:$y \in \displaystyle \bigcap_{x \mathop \in \R_{> 0} } A_x$
+So by definition of [[Definition:Subset|subset]]:
+:$\displaystyle \R_{> 0} \subseteq \bigcap_{x \mathop \in \R_{> 0} } A_x$
+{{qed|lemma}}
+By definition of [[Definition:Set Equality/Definition 2|set equality]]:
+:$\displaystyle \bigcup_{x \mathop \in \R_{> 0} } A_x = \R_{> 0}$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Intersection of Closed Intervals of Positive Reals is Zero}
+Tags: Real Intervals
+
+\begin{theorem}
+Let $\R_{> 0}$ be the [[Definition:Set|set]] of [[Definition:Strictly Positive Real Number|strictly positive real numbers]].
+For all $x \in \R_{> 0}$, let $B_x$ be the [[Definition:Closed Real Interval|closed real interval]] $\closedint 0 x$.
+Then:
+:$\displaystyle \bigcap_{x \mathop \in \R_{> 0} } B_x = \set 0$
+\end{theorem}
+
+\begin{proof}
+Let $\displaystyle B = \bigcap_{x \mathop \in \R_{> 0} } B_x$.
+We have that:
+:$\forall x \in \R_{> 0}: 0 \in \closedint 0 x$
+So by definition of [[Definition:Intersection of Family|intersection]]:
+:$0 \in B$
+and so by [[Singleton of Element is Subset]]:
+:$\displaystyle \set 0 \subseteq \bigcap_{x \mathop \in \R_{> 0} } B_x$
+{{AimForCont}} $\exists y \in \R_{> 0}: y \in B$.
+By definition of $B_x$:
+:$y \notin \closedint 0 {\dfrac y 2 = B_{y/2}$
+and so by definition of [[Definition:Intersection of Family|intersection of family]]:
+:$y \notin B$
+From this [[Proof by Contradiction|contradiction]] it follows that there can be no elements in $B$ apart from $0$.
+That is:
+:$\displaystyle \bigcap_{x \mathop \in \R_{> 0} } A_x \subseteq \set 0$
+By definition of [[Definition:Set Equality/Definition 2|set equality]]:
+:$\displaystyle \bigcap_{x \mathop \in \R_{> 0} } B_x = \set 0$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Union of Closed Intervals of Positive Reals is Set of Positive Reals}
+Tags: Real Intervals
+
+\begin{theorem}
+Let $\R_{> 0}$ be the [[Definition:Set|set]] of [[Definition:Strictly Positive Real Number|strictly positive real numbers]].
+For all $x \in \R_{> 0}$, let $B_x$ be the [[Definition:Closed Real Interval|closed real interval]] $\closedint 0 x$.
+Then:
+:$\displaystyle \bigcup_{x \mathop \in \R_{> 0} } B_x = \R_{\ge 0}$
+\end{theorem}
+
+\begin{proof}
+Let $\displaystyle B = \bigcap_{x \mathop \in \R_{> 0} } B_x$.
+Let $y \in B$.
+Then by definition of [[Definition:Union of Family|union of family]]:
+:$\exists x \in \R_{> 0}: y \in B_x$
+As $B_x \subseteq \R_{> 0}$ it follows by definition of [[Definition:Subset|subset]] that:
+:$y = 0$
+or
+:$y \in \R_{> 0}$
+In either case, $y \in \R_{\ge 0}$
+So:
+:$\displaystyle \bigcap_{x \mathop \in \R_{> 0} } B_x \subseteq \R_{\ge 0}$
+{{qed|lemma}}
+Let $y \in \R_{\ge 0}$.
+If $y = 0$ then $y \in \R_{\ge 0}$ by definition.
+Otherwise, by the [[Archimedean Property]]: 
+:$\exists z \in \N: z > y$
+and so:
+:$y \in B_z$
+That is by definition of [[Definition:Union of Family|union of family]]:
+:$\displaystyle y \in \bigcap_{x \mathop \in \R_{\ge 0} } B_x$
+So by definition of [[Definition:Subset|subset]]:
+:$\displaystyle \R_{\ge 0} \subseteq \bigcap_{x \mathop \in \R_{> 0} } B_x$
+{{qed|lemma}}
+By definition of [[Definition:Set Equality/Definition 2|set equality]]:
+:$\displaystyle \bigcup_{x \mathop \in \R_{> 0} } B_x = \R_{\ge 0}$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Relative Complement of Cartesian Product}
+Tags: Cartesian Product
+
+\begin{theorem}
+Let $A$ and $B$ be [[Definition:Set|sets]].
+Let $X \subseteq A$ and $Y \subseteq B$.
+Then:
+:$\relcomp {A \mathop \times B} {X \times Y} = \paren {A \times \relcomp B Y} \cup \paren {\relcomp A X \times B}$
+\end{theorem}
+
+\begin{proof}
+From [[Set with Relative Complement forms Partition]]:
+:$A = \set {X \mid \relcomp A X}$
+:$B = \set {Y \mid \relcomp B Y}$
+and so by definition of [[Definition:Partition (Set Theory)|partition]]:
+:$A = X \cup \relcomp A X$
+:$B = Y \cup \relcomp B Y$
+By [[Cartesian Product of Unions]]:
+:$A \times B = \paren {X \times Y} \cup \paren {\relcomp A X \times \relcomp B Y} \cup \paren {X \times \relcomp B Y} \cup \paren {\relcomp A X \times Y}$
+and so:
+{{begin-eqn}}
+{{eqn | l = \paren {A \times B} \setminus \paren {X \times Y}
+      | r = \paren {\relcomp A X \times \relcomp B Y} \cup \paren {X \times \relcomp B Y} \cup \paren {\relcomp A X \times Y}
+      | c = 
+}}
+{{eqn | r = \paren {X \times \relcomp B Y} \cup \paren {\relcomp A X \times \relcomp B Y} \cup \paren {\relcomp A X \times Y} \cup \paren {\relcomp A X \times \relcomp B Y}
+      | c = 
+}}
+{{eqn | r = \paren {\paren {X \cup \relcomp A X} \times \relcomp B Y} \cup \paren {\relcomp A X \times \paren {Y \cup \relcomp B Y} }
+      | c = 
+}}
+{{eqn | r = \paren {A \times \relcomp B Y} \cup \paren {\relcomp A X \times B}
+      | c = 
+}}
+{{end-eqn}}
+{{qed}}
+{{MissingLinks|Justification for the steps above}}
+\end{proof}<|endoftext|>
+\section{Subset of Cartesian Product not necessarily Cartesian Product of Subsets}
+Tags: Cartesian Product
+
+\begin{theorem}
+Let $A$ and $B$ be [[Definition:Set|sets]].
+Let $A$ and $B$ both have at least two [[Definition:Distinct|distinct]] [[Definition:Element|elements]].
+Then there exists $W \subseteq A \times B$ such that $W$ is not the [[Definition:Cartesian Product|cartesian product]] of a [[Definition:Subset|subset]] of $A$ and a [[Definition:Subset|subset]] of $B$.
+\end{theorem}
+
+\begin{proof}
+Let $a \in A, b \in A, c \in B, d \in B$ be arbitrary elements of $A$ and $B$.
+Let:
+:$W = \set {\tuple {a, c}, \tuple {a, d}, \tuple {b, d} }$
+Then $W \subseteq A \times B$.
+Suppose $W = X \times Y$ such that $X \subseteq A, Y \subseteq B$.
+Then $a, b \in X$ and $c, d \in Y$.
+But $X \times Y$ also contains $\tuple {b, c}$ which is not in $W$.
+Hence the result.
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Subset equals Preimage of Image implies Injection}
+Tags: Injections, Subsets, Subset equals Preimage of Image implies Injection
+
+\begin{theorem}
+Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
+Let $f^\to: \powerset S \to \powerset T$ be the [[Definition:Direct Image Mapping of Mapping|direct image mapping]] of $f$.
+Similarly, let $f^\gets: \powerset T \to \powerset S$ be the [[Definition:Inverse Image Mapping of Mapping|inverse image mapping]] of $f$.
+Let:
+:$\forall A \in \powerset S: A = \map {\paren {f^\gets \circ f^\to} } A$
+Then $f$ is an [[Definition:Injection|injection]].
+\end{theorem}
+
+\begin{proof}
+Let $f$ be such that:
+:$\forall A \in \powerset S: A = \map {\paren {f^\gets \circ f^\to} } A$
+In particular, it holds for all [[Definition:Subset|subsets]] of $A$ which are [[Definition:Singleton|singletons]].
+Now, consider any $x, y \in A$.
+We have:
+{{begin-eqn}}
+{{eqn | l = \map f x
+      | r = \map f y
+      | c =
+}}
+{{eqn | ll= \leadsto
+      | l = \set {\map f x}
+      | r = \set {\map f y}
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = \map {f^\to} {\set x}
+      | r = \map {f^\to} {\set y}
+      | c = {{Defof|Direct Image Mapping of Mapping}}
+}}
+{{eqn | ll= \leadsto
+      | l = \set x
+      | r = \map {f^\gets} {\map {f^\to} {\set x} }
+      | c = by hypothesis: $A = \map {\paren {f^\gets \circ f^\to} } A$
+}}
+{{eqn | r = \map {f^\gets} {\map {f^\to} {\set y} }
+      | c = 
+}}
+{{eqn | r = \set y
+      | c = by hypothesis: $A = \map {\paren {f^\gets \circ f^\to} } A$
+}}
+{{eqn | ll= \leadsto
+      | l = x
+      | r = y
+      | c = 
+}}
+{{end-eqn}}
+So $f$ is an [[Definition:Injection|injection]].
+{{qed}}
+\end{proof}
+
+\begin{proof}
+Suppose that $f$ is not an [[Definition:Injection|injection]].
+Then two [[Definition:Element|elements]] of $S$ map to the same one [[Definition:Element|element]] of $T$.
+That is:
+:$\exists a_1, a_2 \in S, b \in T: \map f {a_1} = \map f {a_2} = b$
+Let $A = \set {a_1}$.
+Then:
+{{begin-eqn}}
+{{eqn | l = \map {f^\to} A
+      | r = \set b
+      | c = 
+}}
+{{eqn | ll= \leadsto
+      | l = \map {f^\gets} {\map {f^\to} A}
+      | o = \supseteq
+      | r = \set {a_1, a_2}
+      | c = {{Defof|Preimage of Subset under Mapping}}
+}}
+{{eqn | ll= \leadsto
+      | l = \map {f^\gets} {\map {f^\to} A}
+      | o = \supsetneqq
+      | r = A
+      | c = as $A = \set {a_1}$
+}}
+{{end-eqn}}
+So by the [[Rule of Transposition]]:
+:$\forall A \in \powerset S: A = \map {\paren {f^\gets \circ f^\to} } A$
+implies that $f$ is an [[Definition:Injection|injection]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Preimage of Image of Subset under Injection equals Subset}
+Tags: Injections, Composite Mappings, Preimages under Mappings
+
+\begin{theorem}
+Let $f: S \to T$ be an [[Definition:Injection|injection]].
+Then:
+:$\forall A \subseteq S: A = \paren {f^{-1} \circ f} \sqbrk A$
+where:
+:$f \sqbrk A$ denotes the [[Definition:Image of Subset under Mapping|image of $A$ under $f$]]
+:$f^{-1}$ denotes the [[Definition:Inverse of Mapping|inverse]] of $f$
+:$f^{-1} \circ f$ denotes [[Definition:Composition of Mappings|composition]] of $f^{-1}$ and $f$.
+\end{theorem}
+
+\begin{proof}
+Let $f$ be an [[Definition:Injection|injection]].
+From [[Subset of Domain is Subset of Preimage of Image]], we have that:
+:$\forall A \subseteq S: A \subseteq \paren {f^{-1} \circ f} \sqbrk A$
+by dint of $f$ being a [[Definition:Relation|relation]].
+So what we need to do is show that:
+:$\forall A \subseteq S: \paren {f^{-1} \circ f} \sqbrk A \subseteq A$
+Take any $A \subseteq S$.
+Let $x \in A$.
+We have:
+{{begin-eqn}}
+{{eqn | l = x
+      | o = \in
+      | r = \paren {f^{-1} \circ f} \sqbrk A
+      | c = 
+}}
+{{eqn | lll=\leadsto
+      | l = x
+      | o = \in
+      | r = f^{-1} \sqbrk {f \sqbrk A}
+      | c = {{Defof|Composition of Mappings}}
+}}
+{{eqn | lll=\leadsto
+      | l = \map f x
+      | o = \in
+      | r = f \sqbrk A
+      | c = {{Defof|Inverse of Mapping}}
+}}
+{{eqn | lll=\leadsto
+      | ll= \exists y \in A:
+      | l = \map f x
+      | r = \map f y
+      | c = {{Defof|Image of Subset under Mapping}}
+}}
+{{eqn | lll=\leadsto
+      | l = x
+      | r = y
+      | c = {{Defof|Injection}}
+}}
+{{eqn | lll=\leadsto
+      | l = x
+      | o = \in
+      | r = A
+      | c = as $y \in A$
+}}
+{{end-eqn}}
+Thus we see that:
+:$\paren {f^{-1} \circ f} \sqbrk A \subseteq A$
+and hence the result:
+:$\forall A \subseteq S: A = \paren {f^{-1} \circ f} \sqbrk A$
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Subset equals Image of Preimage implies Surjection}
+Tags: Surjections, Subset equals Image of Preimage implies Surjection
+
+\begin{theorem}
+Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
+Let:
+:$\forall B \subseteq T: B = \paren {f \circ f^{-1} } \sqbrk B$
+where $f \sqbrk B$ denotes the [[Definition:Image of Subset under Mapping|image of $B$ under $f$]].
+Then $f$ is a [[Definition:Surjection|surjection]].
+\end{theorem}
+
+\begin{proof}
+Let $f$ be such that:
+:$\forall B \subseteq T: B = \paren {f \circ f^{-1} } \sqbrk B$
+In particular, it holds for $T$ itself.
+Hence:
+{{begin-eqn}}
+{{eqn | l = T
+      | r = \paren {f \circ f^{-1} } \sqbrk T
+      | c = 
+}}
+{{eqn | l = T
+      | r = f \sqbrk {f^{-1} \sqbrk T}
+      | c = {{Defof|Composition of Mappings}}
+}}
+{{eqn | o = \subseteq
+      | r = f \sqbrk S
+      | c = [[Image of Subset under Relation is Subset of Image/Corollary 2|Image of Subset under Relation is Subset of Image: Corollary 2]]
+}}
+{{eqn | r = \Img f
+      | c = {{Defof|Image of Mapping}}
+}}
+{{eqn | o = \subseteq
+      | r = T
+      | c = [[Image is Subset of Codomain/Corollary 1|Image is Subset of Codomain: Corollary 1]]
+}}
+{{end-eqn}}
+So:
+:$T \subseteq \Img f \subseteq T$
+and so by definition of [[Definition:Set Equality/Definition 2|set equality]]:
+:$\Img f = T$
+So, by definition, $f$ is a [[Definition:Surjection|surjection]].
+{{qed}}
+\end{proof}
+
+\begin{proof}
+Suppose $f$ is not a [[Definition:Surjection|surjection]].
+$T$ must have at least two [[Definition:Element|elements]] for this to be the case.
+Let one of these two [[Definition:Element|elements]] not be the [[Definition:Image of Element under Mapping|image]] of any [[Definition:Element|element]] of $S$.
+That is, let $b_1, b_2 \in T$ such that:
+:$\exists a \in S: f \paren a = b_1$
+:$\nexists x \in S: f \paren x = b_2$
+Let $B = \set {b_1, b_2}$.
+Then:
+{{begin-eqn}}
+{{eqn | l = f^{-1} \sqbrk B
+      | r = A
+      | c = for some $A \subseteq S$
+}}
+{{eqn | ll= \leadsto
+      | l = f \sqbrk {f^{-1} \sqbrk B}
+      | r = \set {b_1}
+      | c = as $b_2$ has no [[Definition:Preimage of Element under Mapping|preimage]]
+}}
+{{eqn | ll= \leadsto
+      | l = f \sqbrk {f^{-1} \sqbrk B}
+      | o = \ne
+      | r = B
+      | c = as $B = \set {b_1, b_2}$
+}}
+{{end-eqn}}
+So by the [[Rule of Transposition]]:
+:$\forall B \subseteq S: B = \paren {f \circ f^{-1} } \sqbrk B$
+implies that $f$ is an [[Definition:Surjection|surjection]].
+{{qed}}
+\end{proof}<|endoftext|>
+\section{Image of Preimage of Subset under Surjection equals Subset}
+Tags: Surjections, Preimages under Mappings, Composite Mappings
+
+\begin{theorem}
+Let $f: S \to T$ be a [[Definition:Surjection|surjection]].
+Then:
+:$\forall B \subseteq T: B = \left({f \circ f^{-1} }\right) \sqbrk B$
+where:
+:$f \sqbrk B$ denotes the [[Definition:Image of Subset under Mapping|image of $B$ under $f$]]
+:$f^{-1}$ denotes the [[Definition:Inverse of Mapping|inverse]] of $f$
+:$f \circ f^{-1}$ denotes [[Definition:Composition of Mappings|composition]] of $f$ and $f^{-1}$.
+\end{theorem}
+
+\begin{proof}
+Let $g$ be a [[Definition:Surjection|surjection]].
+Let $B \subseteq T$.
+Let $b \in B$.
+Then:
+{{begin-eqn}}
+{{eqn | ll= \exists a \in S:
+      | l = b
+      | r = \map f a
+      | c = {{Defof|Surjection}}
+}}
+{{eqn | lll=\leadsto
+      | l = a 
+      | o = \in
+      | r = f^{-1} \sqbrk B
+      | c = {{Defof|Preimage of Subset under Mapping}}
+}}
+{{eqn | lll=\leadsto
+      | l = b
+      | o = \in
+      | r = f \sqbrk {f^{-1} \sqbrk B}
+      | c = {{Defof|Image of Subset under Mapping}}
+}}
+{{eqn | lll=\leadsto
+      | l = B
+      | o = \subseteq
+      | r = f \sqbrk {f^{-1} \sqbrk B}
+      | c = {{Defof|Subset}}
+}}
+{{eqn | lll=\leadsto
+      | l = B
+      | o = \subseteq
+      | r = \paren {f \circ f^{-1} } \sqbrk B
+      | c = {{Defof|Composition of Mappings}}
+}}
+{{end-eqn}}
+From [[Subset of Codomain is Superset of Image of Preimage]], we already have that:
+:$\paren {f \circ f^{-1} } \sqbrk B \subseteq B$
+So:
+:$B \subseteq \paren {f \circ f^{-1} } \sqbrk B \subseteq B$
+and by definition of [[Definition:Set Equality/Definition 2|set equality]]:
+:$B = \paren {f \circ f^{-1} } \sqbrk B$
+{{qed}}
+\end{proof}
\ No newline at end of file