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 $y = z$?

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    $\begingroup$ He just used the assumption that $x +y = x + z$. That's why he used associativity (shifted the brackets) in the first place. $\endgroup$ – Student Feb 6 '17 at 11:05
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    $\begingroup$ 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.) $\endgroup$ – Kyle Strand Feb 6 '17 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 $x+y = x+z$.

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