I'm looking at basic proofs in Lang and in Downing. Here's the proof in question in Downing:
$m$ and $n$ are natural numbers $\ne 0$.
$$s = (2n) + (2m + 1)$$ $$s = 2(m + n) + 1$$ Thus the answer is odd by an earlier definition of odd as $2n + 1$. This seems to me like just selectively applying the rules to get an answer you want. Why can't I just rewrite this as $s = 2(m + n + 1)$ by associativity, thus producing an even number? I suspect I'm misunderstanding something, but I can't figure it out.