intros a b nZb p q r H' H'0 H'1 H'2; try assumption.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := spminusf a b nZb
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM)
ltM_dec p
(mults (A:=A) multA (n:=n)
(invTerm (A:=A) invA (n:=n) (T1 A1 n)) q)) r);
auto.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(spminusf a b nZb p r)
(mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n))
q)); auto.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); auto.
Qed.

Theorem spminusf_plusTerm :
forall (a b c : Term A n) (nZc : ~ zeroP (A:=A) A0 eqA (n:=n) c)
(p q r : list (Term A n)),
canonical A0 eqA ltM p ->
canonical A0 eqA ltM q ->
canonical A0 eqA ltM r ->
divP A A0 eqA multA divA n a c ->
divP A A0 eqA multA divA n b c ->
eqT a b ->
~ zeroP (A:=A) A0 eqA (n:=n) (plusTerm (A:=A) plusA (n:=n) a b) ->
eqP A eqA n
(spminusf (plusTerm (A:=A) plusA (n:=n) a b) c nZc
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec p
q) r)
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(spminusf a c nZc p r) (spminusf b c nZc q r)).