Euclid's Lemma for Prime Divisors/Proof 1

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
We have that the integers form a Euclidean domain.

Then from Irreducible Elements of Ring of Integers we have that the irreducible elements of $\Z$ are the primes and their negatives.

The result then follows directly from Euclid's Lemma for Irreducible Elements.