intros a b nZa nZb p q r Cpxa Cpxb H'1.
cut (~ zeroP (A:=A) A0 eqA (n:=n) (ppc (A:=A) A1 (n:=n) a b));
[ intros nZppab | auto ].
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := mults (A:=A) multA (n:=n) (T1 A1 n)
(spolyf (pX a p) (pX b q) Cpxa Cpxb)).
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := mults (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n)
(ppc (A:=A) A1 (n:=n) a b) (b:=ppc (A:=A) A1 (n:=n) a b)
nZppab) (spolyf (pX a p) (pX b q) Cpxa Cpxb)).
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
apply eqTerm_spolyf_red2; auto.
Qed.

(* Specifications: *)
Theorem spoly_is_minus :
forall (a b : Term A n) (nZa : ~ zeroP (A:=A) A0 eqA (n:=n) a)
(nZb : ~ zeroP (A:=A) A0 eqA (n:=n) b) (p q : list (Term A n))
(Cpxa : canonical A0 eqA ltM (pX a p))
(Cpxb : canonical A0 eqA ltM (pX b q)),
eqP A eqA n (spolyf (pX a p) (pX b q) Cpxa Cpxb)
(minuspf A A0 A1 eqA invA minusA multA eqA_dec n ltM ltM_dec
(mults (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n)
(ppc (A:=A) A1 (n:=n) a b) (b:=a) nZa)
(pX a p))
(mults (A:=A) multA (n:=n)
(divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n)
(ppc (A:=A) A1 (n:=n) a b) (b:=b) nZb)
(pX b q))).