Record filter_to_net_DS_set : Type := {
  ftn_S : Ensemble (point_set X);
  ftn_x : point_set X;
  ftn_S_in_F : In (filter_family F) ftn_S;
  ftn_x_in_S : In ftn_S ftn_x
}.Record filter_to_net_DS_set : Type := {
  ftn_S : Ensemble (point_set X);
  ftn_x : point_set X;
  ftn_S_in_F : In (filter_family F) ftn_S;
  ftn_x_in_S : In ftn_S ftn_x
}.Record filter_to_net_DS_set : Type := {
  ftn_S : Ensemble (point_set X);
  ftn_x : point_set X;
  ftn_S_in_F : In (filter_family F) ftn_S;
  ftn_x_in_S : In ftn_S ftn_x
}.Record filter_to_net_DS_set : Type := {
  ftn_S : Ensemble (point_set X);
  ftn_x : point_set X;
  ftn_S_in_F : In (filter_family F) ftn_S;
  ftn_x_in_S : In ftn_S ftn_x
}.Record filter_to_net_DS_set : Type := {
  ftn_S : Ensemble (point_set X);
  ftn_x : point_set X;
  ftn_S_in_F : In (filter_family F) ftn_S;
  ftn_x_in_S : In ftn_S ftn_x
}.Definition filter_to_net_DS : DirectedSet.Definition filter_to_net : Net filter_to_net_DS X :=
  ftn_x.

Lemma filter_cluster_point_impl_filter_to_net_cluster_point: forall x0:point_set X, filter_cluster_point F x0 -> net_cluster_point filter_to_net x0.