
theorem
  for X be set holds Bottom (InclPoset Filt BoolePoset X) = {X}
proof
  let X be set;
  set L = InclPoset Filt BoolePoset X;
  {X} is Filter of BoolePoset X by Th12;
  then {X} in the set of all  Z where Z is Filter of BoolePoset X;
  then {X} in the carrier of RelStr(#Filt BoolePoset X, RelIncl (Filt
    BoolePoset X)#) by WAYBEL_0:def 24;
  then reconsider bX = {X} as Element of L by YELLOW_1:def 1;
A1: for b be Element of L st b is_>=_than {} holds bX <= b
  proof
    let b be Element of L;
    assume b is_>=_than {};
    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 consider b1 be Filter of BoolePoset X such that
A2: b1 = b;
    consider d be object such that
A3: d in b1 by XBOOLE_0:def 1;
    reconsider d as set by TARSKI:1;
A4: d c= X by A3,WAYBEL_8:26;
    now
      let c be object;
      assume c in {X};
      then c = X by TARSKI:def 1;
      hence c in b by A2,A3,A4,WAYBEL_7:7;
    end;
    then {X} c= b;
    hence thesis by YELLOW_1:3;
  end;
  bX is_>=_than {} by YELLOW_0:6;
  then "\/"({},L) = {X} by A1,YELLOW_0:30;
  hence thesis by YELLOW_0:def 11;
end;
