intros Q a q nZa Op0 in0 r; elim r; clear r.
simpl in |- *; intros H' H'0 p H'1 H'2; auto.
apply Rstar_0; auto.
intros a0 r1 Rec Op1 divp0 p Op2 incl0.
cut (canonical A0 eqA ltM r1);
[ intros Cr1 | apply canonical_imp_canonical with (a := a0); auto ].
cut (canonical A0 eqA ltM (pX a q));
[ intros Caq | apply inPolySet_imp_canonical with (L := Q); auto ].
elim divp0; intros H' H'0; clear divp0.
apply reduceplus_trans with (1 := cs) (y := Rminus a nZa p (pX a q) r1); auto.
apply Rec; auto.
red in |- *; intros p' H; apply incl0; auto with datatypes.
apply
reduceplus_eqp_com
with
(1 := cs)
(p := Rminus a nZa p (pX a q) r1)
(q := minuspf A A0 A1 eqA invA minusA multA eqA_dec n ltM ltM_dec
(Rminus a nZa p (pX a q) r1)
(mults (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n) a0 (b:=a) nZa)
(pX a q))); auto.
apply reduce_imp_reduceplus with (1 := cs); auto.
apply minus_is_reduce; auto.
apply Rminus_in; auto with datatypes.
Qed.
Definition Dmult :
forall (a : Term A n) (nZa : ~ zeroP (A:=A) A0 eqA (n:=n) a),
list (Term A n) -> list (Term A n) -> list (Term A n).
intros a nZa p q; elim p; clear p.
exact (pO A n).
intros a1 p1 rec;
exact
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(divpf a nZa (mults (A:=A) multA (n:=n) a1 q)) rec).
Defined.

Theorem canonical_Dmult :
forall (a : Term A n) (nZa : ~ zeroP (A:=A) A0 eqA (n:=n) a)
(p q : list (Term A n)),
canonical A0 eqA ltM p ->
canonical A0 eqA ltM q -> canonical A0 eqA ltM (Dmult a nZa p q).