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Any ideas to show that in integers, , without division algorithm and without the corresponding property in natural numbers? (Note: z is nonzero.)

In my course we are doing one of those "ground-up" constructions, so we started with a few axioms (Peano?), defined addition and multiplication of natural numbers, put the equivalence relation if on , then defined addition and multiplication of the resulting equivalence classes in what I assume is the standard way of doing that here, but its late and I'm tired so I don't want to type anymore. Maybe I'll have time to edit in a few hours after a nap.

My attempt dead-ends me. I compute xz and yz in terms of the equivalence relation definitions (so x, y, and z are equivalence classes as above). I need to show that because xz=yz, it must be true that a+d = b+c (where [(a,b)]=x and [(c,d)]=y). Best I've been able to do so far is a few lines of manipulation that almost seems just random.

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Got it. Had to do a li'l lemma first, which is what I was trying to avoid but gave up and did it another way.