Euclid's Lemma for Euclidean Domains

Theorem
Let $\struct {D, +, \times}$ be a Euclidean domain whose unity is $1$.

Let $a, b, c \in D$.

Let $a \divides b \times c$, where $\divides$ denotes divisibility.

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

Then $a \divides c$.

Also see

 * Euclid's Lemma for Irreducible Elements


 * Euclid's Lemma, for the usual statement of this result, which is this lemma as applied specifically to the integers.