# Problems understanding proof of if $x + y = x + z$ then $y = z$ (Baby Rudin, Chapter 1, Proposition 1.14)

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 $y = z$?

• He just used the assumption that $x +y = x + z$. That's why he used associativity (shifted the brackets) in the first place. Feb 6, 2017 at 11:05
• Note that in any proof, you should expect the original assumption to show up somewhere in the proof; otherwise, it would seem that the proof holds without the original assumption, in which case you've proved a tautology. (I.e. when proving $P \implies Q$, if $P$ is not used in your proof, then you've proved that $Q$ is always true.) Feb 6, 2017 at 19:07

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