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

theorem Th33:
  for X be non empty set, FF be non empty Subset-Family of X st
    FF is Filter of BooleLatt X holds
    FF is Filter of BoolePoset X
  proof
    let X be non empty set,FF be non empty Subset-Family of X such that
A1: FF is Filter of BooleLatt X;
     now
        set Z=LattPOSet BooleLatt X;
        reconsider FF as Subset of Z by A1;
A2:     FF is filtered
        proof
          for x,y being Element of Z st x in FF & y in FF holds
          ex z being Element of Z st z in FF & z <= x & z <= y
          proof
            let x,y be Element of Z such that
A3:         x in FF & y in FF;
            reconsider x1=x,y1=y as Element of BooleLatt X;
            set z = x1"/\" y1;
A4:         now
              x1 /\ y1 c= x1 & x1 /\ y1 c= y1 by XBOOLE_1:17;
              hence z% <= x1% & z% <= y1% by LATTICE3:2,LATTICE3:7;
            end;
            x/\y in FF by A1,A3,Th31;
            hence thesis by A4;
          end;
          hence thesis;
        end;
        FF is upper
        proof
          for x,y be Element of Z st x in FF & x <= y holds y in FF
          proof
            let x,y be Element of Z such that
A5:         x in FF & x <= y;
            reconsider x,y as Element of BooleLatt X;
A6:         x% <= y% by A5;
            x in FF & x c= y by A5,A6,LATTICE3:2,LATTICE3:7;
            hence thesis by A1,Th31;
          end;
          hence thesis;
        end;
        hence thesis by A2;
      end;
      hence thesis;
    end;
