Definition subsgroup_law : law_of_composition H.Definition subsgroup_sgroup : sgroup.Record subsgroup : Type := 
  {subsgroup_part : Predicate G;
   subsgroup_prop :
    forall x y : G,
    in_part x subsgroup_part ->
    in_part y subsgroup_part -> in_part (sgroup_law _ x y) subsgroup_part}.Record subsgroup : Type := 
  {subsgroup_part : Predicate G;
   subsgroup_prop :
    forall x y : G,
    in_part x subsgroup_part ->
    in_part y subsgroup_part -> in_part (sgroup_law _ x y) subsgroup_part}.Record subsgroup : Type := 
  {subsgroup_part : Predicate G;
   subsgroup_prop :
    forall x y : G,
    in_part x subsgroup_part ->
    in_part y subsgroup_part -> in_part (sgroup_law _ x y) subsgroup_part}.Definition sgroup_of_subsgroup (H : subsgroup) :=
  subsgroup_sgroup (subsgroup_prop (s:=H)).

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