Proof of Euclid's Lemma
Without loss of generality, suppose (otherwise we are done). By Bezout's Lemma, there exist integers such that such that . Hence and . Since and (by hypothesis), , as desired.
On the other hand, if is not prime, then it must be composite, i.e., , for integers both greater than 1. Then and . Thus the lemma's converse holds as well.