
theorem
  for X be set holds InclPoset Filt BoolePoset X is upper-bounded
proof
  let X be set;
  set L = InclPoset Filt BoolePoset X;
  bool X is Filter of BoolePoset X by Th11;
  then bool X in the set of all  Z where Z is Filter of BoolePoset X;
  then bool X in the carrier of RelStr(#Filt BoolePoset X, RelIncl (Filt
    BoolePoset X)#) by WAYBEL_0:def 24;
  then reconsider x = bool X as Element of L by YELLOW_1:def 1;
  now
    let b be Element of L;
    assume b in the carrier of L;
    then b in the carrier of RelStr(#Filt BoolePoset X, RelIncl (Filt
      BoolePoset X)#) by YELLOW_1:def 1;
    then b in the set of all  V where V is Filter of BoolePoset X  by
WAYBEL_0:def 24;
    then ex b1 be Filter of BoolePoset X st b1 = b;
    then b c= the carrier of BoolePoset X;
    then b c= bool X by WAYBEL_7:2;
    hence b <= x by YELLOW_1:3;
  end;
  then x is_>=_than the carrier of L by LATTICE3:def 9;
  hence thesis by YELLOW_0:def 5;
end;
