TITLE: Which field property enables us to multiply on both sides by the same value, while preserving equality?
QUESTION [8 upvotes]: I am currently reading through Rudin's Principles of Mathematical Analysis and I am learning about fields and their properties. Note that this is the initial chapter - I am just starting off.
I was wondering which field property enables us to multiply on both sides of an equation and still preserve equality.
There is a very clear proposition stated in the book that gives me this for inequalities:
$$
\text{If} \ \ x > 0 \ \ \text{and} \ \ y < z \ \ \text{then} \ \ xy<xz.
$$
However, the only proposition that seems useful for this in the case of equalities, is stated as an implication and not an equivalence:
$$
\text{If} \ x\not= 0 \ \ \text{and} \ \ xy=xz \ \ \text{then} \ \ y=z.
$$
Any help would be much appreciated.

REPLY [10 votes]: Well, actually this is due to some field property (but one which is usually in the preamble of the definition and not in the list of axioms): The definition of a field states that multiplication is a map $mult$ taking two arguments of the ground set to another one.
And one of the inherent properties of maps is that they have a single output for any given input. That means that if $a=b$ than $mult(m,a)$ and $mult(m,b)$ have the same input and thus, their outputs $ma$ and $mb$ are the same.