Euclid's Lemma/Proof 1

Theorem
Let $a, b, c \in \Z$.

Let $a \mathop \backslash b c$, where $\backslash$ denotes divisibility.

Let $a \perp b$, where $\perp$ denotes relative primeness.

Then $a \mathop \backslash c$.

Proof
Follows directly from Integers are Euclidean Domain.

Also see

 * Euclid's Lemma for Prime Divisors