unfold rel_prime.
split;intro.
apply bezout;trivial.
elim H;intro u;intro H0.
elim H0;intro v;intro.
unfold is_gcd;unfold is_cd.
split.
split;apply one_min_div.
intros.
elim H2;intros.
elim H3;intro q;intro.
elim H4;intro q';intro.
rewrite H5 in H1;rewrite H6 in H1.
case H1;intro.
exists (q*u-q'*v);rewrite mult_minus_lemma2;rewrite mult_assoc;rewrite mult_assoc;trivial.
exists (q'*v-q*u);rewrite mult_minus_lemma2;rewrite mult_assoc;rewrite mult_assoc;trivial.
Qed.

(** gcd(n*a,n*b) = n*gcd(a,b) *)
Lemma gcd_mult : forall (d a b:nat),(is_gcd d a b)->(forall (n:nat),(is_gcd (n*d) (n*a) (n*b))).
unfold is_gcd;unfold is_cd.