Product of Integral Multiples/Proof 2

Proof
First let $m = 0$ or $n = 0$.

, let $m = 0$.

The case where $n = 0$ follows the same lines.

We have:

Next let $m > 0$ and $n > 0$.

The results for $m < 0$ and $n < 0$ follow directly from Powers of Group Elements.