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:

 $\displaystyle \paren {n + \paren {-m} } + m$ $=$ $\displaystyle n + \paren {\paren {-m} + m}$ $\displaystyle$ $=$ $\displaystyle n$ $\displaystyle$ $=$ $\displaystyle p + m$ $\displaystyle$ $=$ $\displaystyle \paren {n - m} + m$

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

So:

$\forall m, n \in \Z, m \le n: n + \paren {-m} = n - m = n -_\N m$

and the result follows.

$\blacksquare$