reserve X for set;

theorem
  for Y being Subset of BoolePoset X holds sup Y = union Y
proof
  set L = BoolePoset X;
  let Y be Subset of L;
A1: the carrier of L = bool X by LATTICE3:def 1;
  then union Y c= union bool X by ZFMISC_1:77;
  then reconsider Un = union Y as Element of L by A1,ZFMISC_1:81;
A2: now
    let b be Element of L;
    assume
A3: b is_>=_than Y;
    for Z being set st Z in Y holds Z c= b by Th2,A3;
    then Un c= b by ZFMISC_1:76;
    hence Un <= b by Th2;
  end;
  for b being Element of L st b in Y holds b <= Un by Th2,ZFMISC_1:74;
  then Un is_>=_than Y;
  hence thesis by A2,YELLOW_0:30;
end;
