So we know for the outcome of the products that A = B. Take any prime p from the first product A. 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. But the factors are all primes themselves , so also b is prime .
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 .