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

theorem Th34:
  for X be non empty set,F be Filter of BoolePoset X holds
    F is Filter of BooleLatt X
  proof
    let X be non empty set,F be Filter of BoolePoset X;
    now
      let Y1,Y2 be Subset of X;
      hereby
        assume that
A1:     Y1 in F and
A2:     Y2 in F;
        reconsider Z1=Y1,Z2=Y2 as Element of BoolePoset X by LATTICE3:def 1;
A3:     Z1 "/\" Z2 in F by A1,A2,WAYBEL_0:41;
        set W = Z1 "/\" Z2;
        reconsider Z1,Z2 as Element of BooleLatt X;
        thus Y1 /\ Y2 in F by A3,YELLOW_1:17;
      end;
      hereby
        assume that
A4:     Y1 in F and
A5:     Y1 c= Y2;
        reconsider Z1=Y1,Z2=Y2 as Element of BoolePoset X by LATTICE3:def 1;
        Z1 <= Z2 by YELLOW_1:2,A5;
        hence Y2 in F by A4,WAYBEL_0:def 20;
      end;
    end;
    hence thesis by Th32;
  end;
