Qed.
Definition bij_group_quo_ker_coKer :
Hom (group_quo G (Ker f) (kernel_normal (G:=G) (G':=G') (f:=f)):GROUP)
(coKer f).
apply
(BUILD_HOM_GROUP
(G:=group_quo G (Ker f) (kernel_normal (G:=G) (G':=G') (f:=f)))
(G':=coKer f)
(ff:=fun x : G =>
Build_subtype (P:=image (sgroup_map (monoid_sgroup_hom f)) (full G))
(subtype_elt:=f x) (coKer_prop f x))).
simpl in |- *.
unfold group_quo_eq, subtype_image_equal in |- *.
simpl in |- *.
intros x y H'; try assumption.
apply
GROUP_reg_right
with (group_inverse _ (Ap (sgroup_map (monoid_sgroup_hom f)) y));
auto with algebra.
apply
Trans
with
(sgroup_law G' (Ap (sgroup_map (monoid_sgroup_hom f)) x)
(Ap (sgroup_map (monoid_sgroup_hom f)) (group_inverse G y)));
auto with algebra.
apply
Trans
with
(Ap (sgroup_map (monoid_sgroup_hom f))
(sgroup_law G x (group_inverse G y))); auto with algebra.
apply Trans with (monoid_unit G'); auto with algebra.
simpl in |- *.
unfold group_quo_eq, subtype_image_equal in |- *.
simpl in |- *.
intros x y; try assumption.
exact (SGROUP_hom_prop f x y).
simpl in |- *.
unfold group_quo_eq, subtype_image_equal in |- *.
simpl in |- *.
auto with algebra.
Defined.

(* Specifications: *)
Lemma bij_group_quo_ker_coKer_bijective : bijective bij_group_quo_ker_coKer.
red in |- *.