reserve x,y,X for set;

theorem
  for T being non empty TopSpace holds T is compact iff for F being
  proper Filter of BoolePoset [#]T ex x being Point of T st x
  is_a_cluster_point_of F, T
proof
  let T be non empty TopSpace;
  hereby
    assume
A1: T is compact;
    let F be proper Filter of BoolePoset [#]T;
    reconsider G = F as proper Filter of BoolePoset [#]T;
    consider x being Point of T such that
A2: x is_a_cluster_point_of a_net G by A1,Lm1;
    take x;
    thus x is_a_cluster_point_of F, T by A2,Th16;
  end;
  assume
A3: for F being proper Filter of BoolePoset [#]T ex x being Point of T
  st x is_a_cluster_point_of F, T;
  now
    let N be net of T such that
    N in NetUniv T;
    reconsider F = a_filter N as proper Filter of BoolePoset the carrier of T;
    consider x being Point of T such that
A4: x is_a_cluster_point_of F, T by A3;
    take x;
    thus x is_a_cluster_point_of N by A4,Th11;
  end;
  hence thesis by Lm2;
end;
