Divisors of Product of Coprime Integers

Theorem
Let $a \mathop \backslash b c$, where $b \perp c$.

Then $a = r s$, where $r \mathop \backslash b$ and $s \mathop \backslash c$.

Proof
Let $r = \gcd \left\{{a, b}\right\}$.

By Divide by GCD for Coprime Integers, $\exists s, t \in \Z: a = r s \land b = r t \land \gcd \left\{{s, t}\right\} = 1$.

So we have written $a = r s$ where $r$ divides $b$.

We now show that $s$ divides $c$.

Since $a$ divides $b c$ there exists $k$ such that $b c = k a$.

Substituting for $a$ and $b$:
 * $r t c = k r s$

which gives:
 * $t c = k s$

So $s$ divides $t c$.

But $s \perp t$ so by Euclid's Lemma $s \mathop \backslash c$ as required.