Subtraction on Integers is Extension of Natural Numbers

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


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 $n + \paren {-m} = n - m$.


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

and the result follows.