Section Lemmas.
Variable R : CRING.
Variable A : algebra R.
intros x x' y y' H' H'0; try assumption.
unfold algebra_mult in |- *.
apply Ap_comp.
auto with algebra.
apply
(Ap_comp (B:=Hom_module (algebra_carrier A) (algebra_carrier A))
(f:=algebra_bilinear_map (algebra_on_def A))
(g:=algebra_bilinear_map (algebra_on_def A)) H').
auto with algebra.
Qed.

Lemma ALGEBRA_lin_right :
forall x y z : A,
Equal (algebra_mult x (sgroup_law A y z))
(sgroup_law A (algebra_mult x y) (algebra_mult x z)).