(** The addition, multiplication and opposite are defined
   in the straightforward way: *)

Definition Qcplus (x y : Qc) := Q2Qc (x+y).Infix "+" := Qcplus : Qc_scope.Definition Qcmult (x y : Qc) := Q2Qc (x*y).Infix "*" := Qcmult : Qc_scope.Definition Qcopp (x : Qc) := Q2Qc (-x).Notation "- x" := (Qcopp x) : Qc_scope.Definition Qcminus (x y : Qc) := x+-y.Infix "-" := Qcminus : Qc_scope.Definition Qcinv (x : Qc) := Q2Qc (/x).Notation "/ x" := (Qcinv x) : Qc_scope.Definition Qcdiv (x y : Qc) := x*/y.Infix "/" := Qcdiv : Qc_scope.

Lemma Q_apart_0_1 : 1 <> 0.