Euclid's Lemma for Prime Divisors/Proof 2

Proof
Let $p \divides a b$.

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:
 * $p \divides b$

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

So:
 * $p \divides a b \implies p \divides a$ or $p \divides b$

as we needed to show.