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

theorem Th36:
  for X be non empty set, F be proper Filter of BoolePoset X holds
  F is Filter of X
  proof
    let X be non empty set,
    F be proper Filter of BoolePoset X;
    A1: F is with_non-empty_elements
    proof
      assume not F is with_non-empty_elements;
      then Bottom BoolePoset X in F by YELLOW_1:18;
      hence contradiction by WAYBEL_7:4;
    end;
    reconsider F as non empty Subset of BooleLatt X;
    thus thesis by A1,Th34,Th35;
  end;
