Original: Some proofs use the fact that if a prime number p divides the product of two natural numbers a and b, then p divides either a or b, a statement known as Euclid's lemma.
Simplified: So we know for the outcome of the products that A = B. Take any prime p from the first product A/O. It divides the A, so it also divides B. Using several times the lemma that we just proved, we can see that p must then divide at least one factor b of B/O. But the factors are all primes themselves, so also b is prime.