intros.
apply Trans with (group_inverse _ (group_inverse _ g)); auto with algebra.
apply GROUP_law_inverse.
apply Trans with (sgroup_law _ (group_inverse _ g') g'); auto with algebra.
Qed.
(* Because of later notation, this should also be known as: *)
Definition min_inj := group_inverse_inj.
(** - Elements of subsets are really pairs of those elements and proofs that they belong to that subset. I prove that two of these pairs are equal (in the subsetoid) iff their first projections are (in the parent setoid) *)

Lemma subtype_elt_comp :
forall (A : Setoid) (B : part_set A) (b b' : B),
Equal b b' -> Equal (subtype_elt b) (subtype_elt b').