Definition Qplus' (p q : Q) := Qred (Qplus p q).
Definition Qmult' (p q : Q) := Qred (Qmult p q).
Definition Qminus' x y := Qred (Qminus x y).

Lemma Qminus'_correct : forall p q : Q, (Qminus' p q)==(Qminus p q).
Add Morphism Qplus' with signature (Qeq ==> Qeq ==> Qeq) as Qplus'_comp.