(** a and b are relatively prime if gcd(a,b)=1 *)
Definition rel_prime (a b:nat) := (is_gcd 1 a b).

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).