Let $a,b$ be natural numbers and $++$ be an increment operation. Based on Peano axioms alone, how to show that if $a=b$, then $a++ = b++$?
Do note that the book I am reading (Real Analysis by Terrence Tao) does not mention that the increment operator is a function (actually at this stage the functions are not even defined yet). I am aware of this post, but it doesn't seem to answer the question, rather than just saying that this is "self-explanatory".
So how would we show it? I'd thought trying induction, but it didn't lead me anywhere.