split; [ try assumption | idtac ].
red in |- *.
simpl in |- *.
unfold group_quo_eq, subtype_image_equal in |- *.
simpl in |- *.
intros x y H'; try assumption.
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
(sgroup_law G' (Ap (sgroup_map (monoid_sgroup_hom f)) x)
(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)
(group_inverse _ (Ap (sgroup_map (monoid_sgroup_hom f)) x)));
auto with algebra.
red in |- *.
intros y; try assumption.
elim y.
intros y' subtype_prf; try assumption.
elim subtype_prf.
intros x H'; try assumption.
elim H'; intros H'0 H'1; try exact H'0; clear H'.
exists x; try assumption.
Qed.

Theorem factor_group_hom :
Equal f
(comp_hom inj_coKer_group
(comp_hom bij_group_quo_ker_coKer surj_group_quo_ker)).