reserve
  X for non empty set,
  FX for Filter of X,
  SFX for Subset-Family of X;

theorem
  for X be non empty set, F be Filter of X,B be basis of F,
  L be Subset of BoolePoset X st L=#B holds F=uparrow L
  proof
    let X be non empty set,F be Filter of X,B be basis of F,
    L be Subset of BoolePoset X;
    assume
A1: L=#B;
    F=<.#B.] by Th06;
    hence F=uparrow L by Th04,A1;
  end;
