intros x y H'; try assumption.
apply GROUP_reg_left with x.
apply Trans with (monoid_unit G); auto with algebra.
Qed.

(* Specifications: *)
Lemma GROUP_inverse_law :
forall x y : G,
Equal (group_inverse _ (sgroup_law _ x y))
(sgroup_law _ (group_inverse _ y) (group_inverse _ x)).