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 euclide_unique : forall (a b q r q' r':nat),(b<>O)->a=b*q+r->a=b*q'+r'->r<b->r'<b->(q=q')/\(r=r').