Record cluster_point_subnet_DS_set : Type := {
  cps_i:DS_set I;
  cps_U:Ensemble (point_set X);
  cps_U_open_neigh: open_neighborhood cps_U x0;
  cps_xi_in_U: In cps_U (x cps_i)
}.Record cluster_point_subnet_DS_set : Type := {
  cps_i:DS_set I;
  cps_U:Ensemble (point_set X);
  cps_U_open_neigh: open_neighborhood cps_U x0;
  cps_xi_in_U: In cps_U (x cps_i)
}.Record cluster_point_subnet_DS_set : Type := {
  cps_i:DS_set I;
  cps_U:Ensemble (point_set X);
  cps_U_open_neigh: open_neighborhood cps_U x0;
  cps_xi_in_U: In cps_U (x cps_i)
}.Record cluster_point_subnet_DS_set : Type := {
  cps_i:DS_set I;
  cps_U:Ensemble (point_set X);
  cps_U_open_neigh: open_neighborhood cps_U x0;
  cps_xi_in_U: In cps_U (x cps_i)
}.Record cluster_point_subnet_DS_set : Type := {
  cps_i:DS_set I;
  cps_U:Ensemble (point_set X);
  cps_U_open_neigh: open_neighborhood cps_U x0;
  cps_xi_in_U: In cps_U (x cps_i)
}.Definition cluster_point_subnet_DS_ord
  (iU1 iU2 : cluster_point_subnet_DS_set) : Prop :=
  DS_ord (cps_i iU1) (cps_i iU2) /\
  Included (cps_U iU2) (cps_U iU1).Definition cluster_point_subnet_DS : DirectedSet.Definition cluster_point_subnet : Net
  cluster_point_subnet_DS X :=
  fun (iU:DS_set cluster_point_subnet_DS) =>
  x (cps_i iU).

Lemma cluster_point_subnet_is_subnet: Subnet cluster_point_subnet.