TITLE: Problems understanding proof of if $x + y = x + z$ then $y = z$ (Baby Rudin, Chapter 1, Proposition 1.14)
QUESTION [20 upvotes]: I'm having trouble with whether Rudin actually proves what he's tried to prove.
Proposition 1.14; (page 6)
The axioms of addition imply the following statements:
a) if $x + y = x + z$ then $y = z$
The author's proof is as follows:
$ y = (0 + y) = (x + -x) + y = -x + (x + \textbf{y})$
$$ = -x + (x + \textbf{z}) = (-x + x) + z = (0 + z) = z $$
I emphased the section which troubles me. 
How does Rudin prove that $ y = z $ if he substituted $y = z$?

REPLY [53 votes]: He didn't substitute $z$ for $y$; rather, he substituted $x+z$ for $x+y$. This is legitimate based on the assumption that $x+y = x+z$.