Definition subgroup_inv : MAP H H.Definition subgroup_group : group.Record subgroup : Type := 
  {subgroup_submonoid : submonoid G;
   subgroup_prop :
    forall x : G,
    in_part x subgroup_submonoid ->
    in_part (group_inverse _ x) subgroup_submonoid}.Record subgroup : Type := 
  {subgroup_submonoid : submonoid G;
   subgroup_prop :
    forall x : G,
    in_part x subgroup_submonoid ->
    in_part (group_inverse _ x) subgroup_submonoid}.Record subgroup : Type := 
  {subgroup_submonoid : submonoid G;
   subgroup_prop :
    forall x : G,
    in_part x subgroup_submonoid ->
    in_part (group_inverse _ x) subgroup_submonoid}.Definition group_of_subgroup (H : subgroup) :=
  subgroup_group (subgroup_prop (s:=H)).

Lemma subgroup_in_prop : forall x : G, in_part x H -> in_part (group_inverse _ x) H.