reserve x,y,X for set;

theorem Th26:
  for T being non empty TopSpace, A being Subset of T for x being
Point of T holds x in Cl A iff ex F being ultra Filter of BoolePoset [#]T st A
  in F & x is_a_convergence_point_of F, T
proof
  let T be non empty TopSpace, A be Subset of T;
  let x be Point of T;
  hereby
    assume x in Cl A;
    then consider N being net of T such that
A1: N is_eventually_in A and
A2: x is_a_cluster_point_of N by Th21;
    consider S being subnet of N such that
A3: x in Lim S by A2,WAYBEL_9:32;
    set F = a_filter S;
    consider G being Filter of BoolePoset [#]T such that
A4: F c= G and
A5: G is ultra by WAYBEL_7:26;
    reconsider G as ultra Filter of BoolePoset [#]T by A5;
    take G;
    S is_eventually_in A by A1,Th19;
    then A in F;
    hence A in G by A4;
    x is_a_convergence_point_of F, T by A3,Th12;
    hence x is_a_convergence_point_of G, T by A4;
  end;
  given F being ultra Filter of BoolePoset [#]T such that
A6: A in F and
A7: x is_a_convergence_point_of F, T;
  x is_a_cluster_point_of F, T by A7,WAYBEL_7:27;
  hence thesis by A6,Th25;
end;
