Euclid's Lemma for Prime Divisors/Proof 2

Lemma
Let $p$ be a prime number.

Let $a$ and $b$ be integers such that:
 * $p \mathop \backslash a b$

where $\backslash$ means is a divisor of.

Then $p \mathop \backslash a$ or $p \mathop \backslash b$.

Proof
Let $p \mathop \backslash a b$.

Suppose $p \nmid a$.

Then from the definition of prime:
 * $p \perp a$

where $\perp$ indicates that $p$ and $a$ are coprime.

Thus from Euclid's Lemma it follows that:
 * $p \mathop \backslash b$

Similarly, if $p \nmid b$ it follows that $p \mathop \backslash a$.

So:
 * $p \mathop \backslash a b \implies p \mathop \backslash a$ or $p \mathop \backslash b$

as we needed to show.