Subtraction on Integers is Extension of Natural Numbers

Theorem
Integer subtraction is an extension of the definition of subtraction on the natural numbers.

Proof

 * Let $m, n \in \N: m \le n$.

From natural number subtraction, $\exists p \in \N: m + p = n$ such that $n - m = p$.

As $m, n, p \in \N$, it follows that $m, n, p \in \Z$ as well.

However, as $\Z$ is the inverse completion of $\N$, it follows that $-m \in \Z$ as well, so it makes sense to express the following:

Thus, as all elements of $\Z$ are cancellable, it follows that $n + \left({-m}\right) = n - m$.

So:
 * $\forall m, n \in \Z, m \le n: n + \left({-m}\right) = n - m = n -_\N m$

and the result follows.