
theorem
  for X be set holds InclPoset Filt BoolePoset X is lower-bounded
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 x = {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 consider b1 be Filter of BoolePoset X such that
A1: b1 = b;
    consider d be object such that
A2: d in b1 by XBOOLE_0:def 1;
    reconsider d as set by TARSKI:1;
A3: d c= X by A2,WAYBEL_8:26;
    now
      let c be object;
      assume c in {X};
      then c = X by TARSKI:def 1;
      hence c in b by A1,A2,A3,WAYBEL_7:7;
    end;
    then {X} c= b;
    hence x <= b by YELLOW_1:3;
  end;
  then x is_<=_than the carrier of L by LATTICE3:def 8;
  hence thesis by YELLOW_0:def 4;
end;
