elim H0;intro q;intro.
replace b with (b*1);try ring.
rewrite H1.
apply mult_le_compat;try omega.
destruct q;omega.
Qed.

(** Euclide theorem (existence) *)
Theorem euclide : forall (a b:nat),(b<>O)->{q:nat & { r:nat | (a=b*q+r) /\ (r < b)}}.
intros.