Existence of Lowest Common Multiple/Proof 3

Proof
Note that as Integer Divides Zero, both $a$ and $b$ are divisors of zero.

Thus by definition $0$ is a common multiple of $a$ and $b$.

Non-trivial common multiples of $a$ and $b$ exist.

Indeed, $a b$ and $-\paren {a b}$ are common multiples of $a$ and $b$.

Either $a b$ or $-\paren {a b}$ is strictly positive.

Let $S$ denote the set of strictly positive common multiples of $a$ and $b$.

By the Well-Ordering Principle, $S$ contains a smallest element.

This can then be referred to as the lowest common multiple of $a$ and $b$.