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 $$m, n \in \mathbb{N}: m \le n$$.

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

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

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

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

So $$\forall m, n \in \mathbb{Z}, m \le n: n + \left({-m}\right) = n - m = n -_\mathbb{N} m$$ and the result follows.