Divisors of Product of Coprime Integers/Corollary

Corollary to Divisors of Product of Coprime Integers
Let $p$ be a prime.

Let $p \mathop \backslash b c$, where $b \perp c$.

Then $p \mathop \backslash b$ or $p \mathop \backslash c$, but not both.

Proof
From the main result, $p = r s$, where $r \mathop \backslash b$ and $s \mathop \backslash c$.

But as $p$ is prime, either:
 * $r = 1$ and $s = p$, or:
 * $r = p$ and $s = 1$.

So $p \mathop \backslash b$ or $p \mathop \backslash c$.

But $p$ can not divide both as $b \perp c$.