simpl in |- *.
intros x y H'; try assumption.
apply
Trans
with
(sgroup_law _ (Ap (sgroup_map (monoid_sgroup_hom f)) x)
(Ap (sgroup_map (monoid_sgroup_hom f))
(sgroup_law G y (group_inverse G x)))); auto with algebra.
apply
Trans
with
(sgroup_law _ (Ap (sgroup_map (monoid_sgroup_hom f)) x)
(sgroup_law _ (Ap (sgroup_map (monoid_sgroup_hom f)) y)
(Ap (sgroup_map (monoid_sgroup_hom f)) (group_inverse G x))));
auto with algebra.
apply
Trans
with
(sgroup_law _ (Ap (sgroup_map (monoid_sgroup_hom f)) x)
(sgroup_law _ (Ap (sgroup_map (monoid_sgroup_hom f)) y)
(group_inverse _ (Ap (sgroup_map (monoid_sgroup_hom f)) x))));
auto with algebra.
apply
Trans
with
(sgroup_law _ (Ap (sgroup_map (monoid_sgroup_hom f)) x)
(sgroup_law _ (monoid_unit G')
(group_inverse _ (Ap (sgroup_map (monoid_sgroup_hom f)) x))));
auto with algebra.
apply
Trans
with
(sgroup_law _ (Ap (sgroup_map (monoid_sgroup_hom f)) x)
(group_inverse _ (Ap (sgroup_map (monoid_sgroup_hom f)) x)));
auto with algebra.
Qed.
Set Strict Implicit.
Unset Implicit Arguments.
Definition group_quo_ker := group_quo G Ker kernel_normal.
Set Implicit Arguments.
Unset Strict Implicit.

Lemma Ker_prop : forall x : G, in_part x Ker -> Equal (f x) (monoid_unit G').
auto with algebra.