Section Def.
Variable G : GROUP.
Variable H : subgroup G.
Definition normal :=
forall x y : G,
in_part y H ->
in_part (sgroup_law _ x (sgroup_law _ y (group_inverse _ x))) H.
Hypothesis Hnormal : normal.
Definition group_quo_eq (x y : G) :=
in_part (sgroup_law _ x (group_inverse _ y)) H.
Definition group_quo_eqrel : Relation G.
apply (Build_Relation (E:=G) (Rel_fun:=group_quo_eq)).
red in |- *.
unfold app_rel, group_quo_eq in |- *.
intros x x' y y' H' H'0 H'1; try assumption.
apply in_part_comp_l with (sgroup_law G x (group_inverse G y));
auto with algebra.
Defined.
red in Hnormal.
red in |- *.
split; [ try assumption | idtac ].
red in |- *.
intros x; red in |- *.
simpl in |- *.
unfold app_rel, group_quo_eq in |- *.
apply
in_part_comp_l
with (sgroup_law G x (sgroup_law G (monoid_unit G) (group_inverse G x)));
auto with algebra.
red in |- *.
split; [ try assumption | idtac ].
red in |- *.
simpl in |- *.
unfold app_rel, group_quo_eq in |- *.
intros x y z H' H'0; try assumption.
apply
in_part_comp_l
with
(sgroup_law G (sgroup_law G x (group_inverse G y))
(sgroup_law G y (group_inverse G z))); auto with algebra.
apply
Trans
with
(sgroup_law G x
(sgroup_law G (group_inverse G y) (sgroup_law G y (group_inverse G z))));
auto with algebra.
apply
Trans
with
(sgroup_law G x
(sgroup_law G (sgroup_law G (group_inverse G y) y) (group_inverse G z)));
auto with algebra.
apply
Trans
with (sgroup_law G x (sgroup_law G (monoid_unit G) (group_inverse G z)));
auto with algebra.
red in |- *.
simpl in |- *.
unfold app_rel, group_quo_eq in |- *.
intros x y H'; try assumption.
apply
in_part_comp_l with (group_inverse G (sgroup_law G x (group_inverse G y)));
auto with algebra.
apply
Trans
with
(sgroup_law G (group_inverse G (group_inverse G y)) (group_inverse G x));
auto with algebra.
Qed.
Definition group_quo_set := quotient G group_quo_eqrel group_quo_eqrel_equiv.

Lemma normal_com_in :
forall x y : G, in_part (sgroup_law _ x y) H -> in_part (sgroup_law _ y x) H.