intros x y H' H'0; try assumption.
apply (subsgroup_prop (G:=G) (s:=H)); auto with algebra.
Qed.
Definition inj_subsgroup : Hom (H:SGROUP) G.
apply (Build_sgroup_hom (E:=H) (F:=G) (sgroup_map:=inj_part H)).
red in |- *.
auto with algebra.
Defined.

(* Specifications: *)
Lemma inj_subgroup_injective : injective inj_subsgroup.
red in |- *.