
theorem
  for X being set, Y being non empty lower Subset of BoolePoset X holds
Y is directed iff for Z being finite Subset-Family of X st Z c= Y holds union Z
  in Y
proof
  let X be set, Y be non empty lower Subset of BoolePoset X;
  hereby
    assume
A1: Y is directed;
    let Z be finite Subset-Family of X;
    reconsider B = Z as Subset of BoolePoset X by Th2;
    assume Z c= Y;
    then reconsider A = Z as finite Subset of Y;
A2: A <> {} implies sup B in Y by A1,WAYBEL_0:42;
    Bottom BoolePoset X in Y by A1,WAYBEL_4:21;
    hence union Z in Y by A2,YELLOW_1:18,21,ZFMISC_1:2;
  end;
  assume
A3: for Z being finite Subset-Family of X st Z c= Y holds union Z in Y;
A4: the carrier of BoolePoset X = bool X by Th2;
  now
    let A be finite Subset of Y;
    reconsider Z = A as finite Subset-Family of X by A4,XBOOLE_1:1;
    assume A <> {};
    reconsider Z as finite Subset-Family of X;
    A c= the carrier of BoolePoset X by XBOOLE_1:1;
    then union Z = "\/"(A, BoolePoset X) by YELLOW_1:21;
    hence "\/"(A, BoolePoset X) in Y by A3;
  end;
  hence thesis by WAYBEL_0:42;
end;
