stylistic:style_transfer, information:filtering, information:highlighting, linguistic:paraphrasing
Uniqueness

Suppose, to the contrary, there is an integer that has two distinct prime factorizations. Let n be the least such integer and write n = p p... p = q q... q, where each p and q is prime. We see that p divides q q... q, so p divides some q by Euclid's lemma. Without loss of generality, say p divides q. Since p and q are both prime, it follows that p = q. Returning to our factorizations of n, we may cancel these two factors to conclude that p... p = q... q. We now have two distinct prime factorizations of some integer strictly smaller than n, which contradicts the minimality of n.