# Subtraction on Integers is Extension of Natural Numbers

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## 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.

$\blacksquare$