reserve x,y,X for set;

theorem Th18:
  for T being non empty TopSpace, x being Point of T, A being
  Subset of T st x in Cl A for F being proper Filter of BoolePoset [#]T st F =
  NeighborhoodSystem x holds a_net F is_often_in A
proof
  let T be non empty TopSpace, x be Point of T, A be Subset of T;
  assume
A1: x in Cl A;
  let F be proper Filter of BoolePoset [#]T such that
A2: F = NeighborhoodSystem x;
  set N = a_net F;
  let i be Element of N;
A3: the carrier of N = {[a, f] where a is Element of T, f is Element of F: a
  in f} by Def4;
  i in the carrier of N;
  then consider a being Element of T, f being Element of F such that
A4: i = [a, f] and
  a in f by A3;
  reconsider f as a_neighborhood of x by A2,Th2;
A5: i`2 = f by A4;
  f meets A by A1,CONNSP_2:27;
  then consider b being object such that
A6: b in f and
A7: b in A by XBOOLE_0:3;
  reconsider b as Element of T by A6;
  [b, f] in the carrier of N by A3,A6;
  then reconsider j = [b, f] as Element of N;
  take j;
A8: j`1 = b;
  j`2 = f;
  hence i <= j & N.j in A by A7,A5,A8,Def4;
end;
