now we will prove that we can write an integer greater than 1 in "only one" way as a product of prime numbers. take two products of primes "a" and "b" which have the same outcome. 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. but we know that "p" is also prime, so "p" must be equal to "b". so now we divide "a" by "p" and also divide "b" by "p". and we get a result like "a*" = "b*". again we can take a prime "p" from the first product "a*" and find out that is equal to some number in the product "b*". continuing in this way, at the end we see that the prime factors of the two products must be exactly the same. this proves that we can write a positive integer as a product of primes in only one unique way.