PID implies UFD

Statement
Any principal ideal domain is a unique factorization domain.

Facts used

 * 1) uses::Principal ideal implies Noetherian
 * 2) uses::Noetherian implies ACCP
 * 3) uses::ACCP implies every element has a factorization into irreducibles
 * 4) uses::Principal ideal implies Bezout
 * 5) uses::Bezout implies every irreducible is prime
 * 6) uses::Every irreducible is prime implies any two irreducible factorizations are equal upto ordering and associates

Proof
Facts (1)-(3) guarantee the existence of a factorization into irreducibles for any nonzero non-unit. Facts (4)-(6) guarantee the uniqueness of factorization.