Fundamental theorem of arithmetic
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.