
theorem Th11:
  for X be set holds bool X is Filter of BoolePoset X
proof
  let X be set;
  bool X c= the carrier of BoolePoset X by WAYBEL_7:2;
  then reconsider A = bool X as non empty Subset of BoolePoset X;
A1: now
    let x,y be set;
    assume x in A & y in A;
    then x /\ y c= X /\ X by XBOOLE_1:27;
    hence x /\ y in A;
  end;
  for x,y be set st x c= y & y c= X & x in A holds y in A;
  then A is upper by WAYBEL_7:7;
  hence thesis by A1,WAYBEL_7:9;
end;
