Section Lemmas.
Variable G : GROUP.
unfold group_inverse in |- *.
auto with algebra.
Qed.
intros; apply (group_inverse_r_prf G x); auto with algebra.
Qed.
intros; apply (group_inverse_l_prf G x); auto with algebra.
Qed.
Hint Resolve GROUP_comp GROUP_inverse_r GROUP_inverse_l: algebra.

Lemma GROUP_unit_inverse :
Equal (group_inverse _ (monoid_unit G)) (monoid_unit G).