Definition normal :=
  forall x y : G,
  in_part y H ->
  in_part (sgroup_law _ x (sgroup_law _ y (group_inverse _ x))) H.Definition group_quo_eq (x y : G) :=
  in_part (sgroup_law _ x (group_inverse _ y)) H.Definition group_quo_eqrel : Relation G.

Lemma group_quo_eqrel_equiv : equivalence group_quo_eqrel.