intros a x; try assumption.
apply Sym.
apply GROUP_law_inverse.
apply Trans with (module_mult a (sgroup_law Mod x (group_inverse Mod x)));
auto with algebra.
apply Trans with (module_mult a (monoid_unit Mod)); auto with algebra.
apply MODULE_absorbant_r.
Qed.

Lemma MODULE_mult_op_l :
forall (a : R) (x : Mod),
Equal (module_mult (group_inverse R a) x)
(group_inverse Mod (module_mult a x)).