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.