intros p; elim p; simpl in |- *; auto.
intros a l H' a0 H'0.
apply
(eqp_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
l (mults (invTerm (A:=A) invA (n:=n) a0) l));
auto.
change
(eqP A eqA n
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec
(pX a l)
(pX (multTerm (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) a0) a)
(mults (invTerm (A:=A) invA (n:=n) a0) l)))
(pluspf (A:=A) A0 (eqA:=eqA) plusA eqA_dec (n:=n) (ltM:=ltM) ltM_dec l
(mults (invTerm (A:=A) invA (n:=n) a0) l)))
in |- *.
apply (eqp_sym _ _ _ _ _ _ _ _ _ cs n); apply pluspf_inv3a with (1 := cs);
auto.
apply (eqT_trans A n) with (y := multTerm (A:=A) multA (n:=n) a0 a); auto.
apply (T1_eqT _ A1 eqA); auto.
apply
zeroP_comp_eqTerm
with
(1 := cs)
(a := plusTerm (A:=A) plusA (n:=n) a (invTerm (A:=A) invA (n:=n) a));
auto.
apply plusTerm_invTerm_zeroP with (1 := cs); auto.
apply plusTerm_comp_r with (1 := cs); auto.
change
(eqT a (multTerm (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) a0) a))
in |- *.
apply (eqT_trans A n) with (y := multTerm (A:=A) multA (n:=n) (T1 A1 n) a);
auto.
apply (T1_eqT _ A1 eqA); auto.
apply multTerm_eqT; auto.
apply (eqT_trans A n) with (y := a0); auto.
apply (eqT_sym A n); auto.
apply (eqTerm_imp_eqT A eqA); auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with
(y := invTerm (A:=A) invA (n:=n)
(multTerm (A:=A) multA (n:=n) a (T1 A1 n)));
auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with (y := invTerm (A:=A) invA (n:=n) (multTerm (A:=A) multA (n:=n) a a0));
auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n);
apply eqTerm_invTerm_comp with (1 := cs); auto.
apply
(eqTerm_trans _ _ _ _ _ _ _ _ _ cs n)
with (y := invTerm (A:=A) invA (n:=n) (multTerm (A:=A) multA (n:=n) a0 a));
auto.
apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n);
apply mult_invTerm_com with (1 := cs); auto.
Qed.

(* Specifications: *)
Theorem mults_multTerm :
forall (p : list (Term A n)) (a b : Term A n),
eqP A eqA n (mults (multTerm (A:=A) multA (n:=n) a b) p)
(mults a (mults b p)).