# Natural Number Subtraction is not Closed

## Theorem

The operation of subtraction on the natural numbers is not closed.

## Proof

By definition of natural number subtraction:

$n - m = p$

where $p \in \N$ such that $n = m + p$.

However, when $m > n$ there exists no $p \in \N$ such that $n = m + p$.

$\blacksquare$