simpl in |- *.
unfold group_quo_eq, subtype_image_equal in |- *.
intros y; exists y; try assumption.
simpl in |- *.
apply Trans with (Ap (sgroup_map (monoid_sgroup_hom f)) (monoid_unit G));
auto with algebra.
Qed.
Definition inj_coKer_group : Hom (coKer f:GROUP) G'.
apply (BUILD_HOM_GROUP (G:=coKer f) (G':=G') (ff:=inj_part (coKer f))).
simpl in |- *.
auto with algebra.
simpl in |- *.
auto with algebra.
simpl in |- *.
auto with algebra.
Defined.

(* Specifications: *)
Lemma inj_coKer_group_injective : injective inj_coKer_group.
exact (inj_part_injective (E:=G') (A:=coKer f)).