intros.
elim H0;intros.
intro.
elim H1;intros.
elim H5;intros.
rewrite H3 in H6;simpl in H6;auto.
Qed.

(** we now exhibit an algorithm that computes Bezout coefficient: for all a b, there is u and v such that a*u-b*v = gcd(a,b) or b*v-a*u = gcd(a,b) *)
(** the 4 lemmae gives the idea of the algorithm *)
Lemma bezout_aux1 : forall (x y u v:nat),(x<=y)->(is_gcd (u*x-v*(y-x)) x (y-x))->(is_gcd ((u+v)*x-v*y) x y).
intros.