intros x y z H'; try assumption.
apply Trans with (sgroup_law _ y (sgroup_law _ x (group_inverse _ x)));
auto with algebra.
apply Trans with (sgroup_law _ y (monoid_unit G)); auto with algebra.
apply Trans with (sgroup_law _ (sgroup_law _ y x) (group_inverse _ x));
auto with algebra.
apply Trans with (sgroup_law _ (sgroup_law _ z x) (group_inverse _ x));
auto with algebra.
apply Trans with (sgroup_law _ z (sgroup_law _ x (group_inverse _ x)));
auto with algebra.
apply Trans with (sgroup_law _ z (monoid_unit G)); auto with algebra.
Qed.

Lemma GROUP_inverse_inverse :
forall x : G, Equal (group_inverse _ (group_inverse _ x)) x.