intros.
generalize (gcd_is_gcd a p);intros.
case (prime_gcd (gcd a p) p a H H1);intro;rewrite H2 in H1;try tauto.
elim H1;intros.
elim H3;trivial.
Qed.

(** if p is prime and p | a*b then p | a or p | b *)
Lemma prime_mult : forall (p a b:nat),(is_prime p)->(divides (a*b) p)->((divides a p)\/(divides b p)).
intros.