unfold rel_prime.
generalize (gcd_is_gcd a b);intro.
case (eq_nat_dec (gcd a b) 1);intro.
left;rewrite e in H;trivial.
right;intro;apply n;apply (gcd_unique (gcd a b) 1 a b);trivial.
Qed.

(* Specifications: *)
(** if gcd(a,b)=1 and gcd(a,c)=1 then gcd(a,b*c)=1 *)
Lemma rel_prime_mult : forall (a b c:nat),(rel_prime a b)->(rel_prime a c)->(rel_prime a (b*c)).
intros.