apply eqA_trans with (y := plusA (invA a) (multA A0 a)); auto.
apply eqA_trans with (y := plusA (invA a) (multA (plusA A1 (invA A1)) a));
auto.
apply
eqA_trans
with (y := plusA (invA a) (plusA (multA A1 a) (multA (invA A1) a)));
auto.
apply plusA_eqA_comp with (1 := cs); auto.
apply eqA_sym; auto.
apply multA_dist_r.
apply eqA_trans with (y := plusA (invA a) (plusA a (multA (invA A1) a)));
auto.
apply eqA_trans with (y := plusA (plusA (invA a) a) (multA (invA A1) a));
auto.
apply eqA_trans with (y := plusA (plusA a (invA a)) (multA (invA A1) a));
auto.
apply eqA_trans with (y := plusA A0 (multA (invA A1) a)); auto.
apply eqA_trans with (y := plusA (multA (invA A1) a) A0); auto.
Qed.

Theorem divA_A0_l : forall (a : A) (nZa : ~ eqA a A0), eqA (divA A0 a nZa) A0.
intros a nZa; apply eqA_trans with (y := divA (multA A0 a) a nZa).