intros a; try assumption.
apply GROUP_reg_left with (module_mult a (monoid_unit (module_carrier Mod)));
auto with algebra.
apply
Trans
with (module_mult a (sgroup_law Mod (monoid_unit Mod) (monoid_unit Mod)));
auto with algebra.
apply Trans with (module_mult a (monoid_unit Mod)); auto with algebra.
Qed.

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