Product of Coprime Factors

Theorem
Let $a, b, c \in \Z$ such that $a$ and $b$ are coprime.

Let both $a$ and $b$ be divisors of $c$.

Then $a b$ is also a divisor of $c$.

That is:
 * $a \perp b \land a \mathop \backslash c \land b \mathop \backslash c \implies a b \mathop \backslash c$

Proof
We have:
 * $a \mathop \backslash c \implies \exists r \in \Z: c = a r$
 * $b \mathop \backslash c \implies \exists s \in \Z: c = b s$

So: