Definition quotient_euclide (a b:nat)(H:(b<>O)) := let (q,_) := (euclide a b H) in q.Definition remainder_euclide (a b:nat)(H:(b<>O)) := let (_,e0) := (euclide a b H) in let (r,_) := e0 in r.

Lemma rem_euclide : forall (a b:nat)(H:(b<>O)),(remainder_euclide a b H)<b.