intros Q p H'; elim H'; auto.
intros q H'0.
cut (canonical A0 eqA ltM q);
[ intros Op1 | apply CombLinear_canonical with (Q := Q); auto ].
apply CombLinear_comp with (p := q); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pO A n) q); auto; apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n);
auto.
intros a p0 q s H'0 H'1 H'2 H'3 H'4 q0 H'5.
cut (canonical A0 eqA ltM q);
[ intros C0 | apply inPolySet_imp_canonical with (L := Q); auto ].
cut (canonical A0 eqA ltM q0);
[ intros Op1 | apply CombLinear_canonical with (Q := Q); auto ].
cut (canonical A0 eqA ltM p0);
[ intros Op2 | apply CombLinear_canonical with (Q := Q); auto ].
apply
CombLinear_1
with
(a := a)
(p := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
p0 q0)
(q := q); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec (mults (A:=A) multA (n:=n) a q) p0) q0);
auto.
apply eqp_pluspf_com with (1 := cs); auto.
apply
eqp_imp_canonical
with
(p := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(mults (A:=A) multA (n:=n) a q) p0)
(1 := cs); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
Qed.
Hint Resolve CombLinear_pluspf.

Theorem CombLinear_mults1 :
forall (Q : list (poly A0 eqA ltM)) (a : Term A n) (p : list (Term A n)),
~ zeroP (A:=A) A0 eqA (n:=n) a ->
CombLinear Q p -> CombLinear Q (mults (A:=A) multA (n:=n) a p).