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

theorem Th37:
  for T being non empty TopSpace,
  F be Filter of the carrier of T
  ex F1 be proper Filter of BoolePoset the carrier of T st
  F=F1
  proof
    let T be non empty TopSpace,
    F be Filter of the carrier of T;
    reconsider F1=F as non empty Subset of BooleLatt the carrier of T
    by LATTICE3:def 1;
    A1: F1 is Filter of BoolePoset the carrier of T by Th33,Th35;
    not {} in F by CARD_FIL:def 1;
    then not Bottom BoolePoset the carrier of T in F1 by YELLOW_1:18;
    then F1 is proper;
    hence thesis by A1;
end;
