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.

Lemma MODULE_dist_r :
forall (a b : R) (x : Mod),
Equal (module_mult (sgroup_law R a b) x)
(sgroup_law Mod (module_mult a x) (module_mult b x)).