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.
intros x y z; try assumption.
unfold algebra_mult in |- *.
auto with algebra.
Qed.
Parameter
ALGEBRA_lin_left :
forall x y z : A,
Equal (algebra_mult (sgroup_law A x y) z)
(sgroup_law A (algebra_mult x z) (algebra_mult y z)).

Lemma ALGEBRA_mult_lin_right :
forall (x y : A) (a : R),
Equal (algebra_mult x (module_mult a y)) (module_mult a (algebra_mult x y)).