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.

Lemma group_quo_eqrel_equiv : equivalence group_quo_eqrel.