Section Lemmas.
Variable R : RING.
Variable Mod : MODULE R.
intros a b x y H' H'0; try assumption.
apply Trans with (module_mult a y); unfold module_mult in |- *.
apply Ap_comp; auto with algebra.
apply Ap_comp; auto with algebra.
Qed.
exact (operation_assoc (module_op Mod)).
Qed.
exact (module_op_lin_left_prf Mod).
Qed.

Lemma MODULE_dist_l :
forall (a : R) (x y : Mod),
Equal (module_mult a (sgroup_law Mod x y))
(sgroup_law Mod (module_mult a x) (module_mult a y)).