reserve x,y,X for set;

theorem Th25:
  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 proper Filter of BoolePoset [#]T st A
  in F & x is_a_cluster_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;
    set F = a_filter N;
    take F;
    thus A in F by A1;
    thus x is_a_cluster_point_of F, T by A2,Th11;
  end;
  given F being proper Filter of BoolePoset [#]T such that
A3: A in F and
A4: x is_a_cluster_point_of F, T;
  reconsider F9 = F as proper Filter of BoolePoset [#]T;
A5: a_filter a_net F9 = F by Th14;
  then
A6: x is_a_cluster_point_of a_net F9 by A4,Th11;
  a_net F9 is_eventually_in A by A3,A5,Th10;
  hence thesis by A6,Th21;
end;
