
theorem
  for X being set, Y being lower Subset of BoolePoset X holds Y is
  directed iff for x,y being set st x in Y & y in Y holds x \/ y in Y
proof
  let X be set, Y be lower Subset of BoolePoset X;
  hereby
    assume
A1: Y is directed;
    let x,y be set;
    assume
A2: x in Y & y in Y;
    then reconsider a = x, b = y as Element of BoolePoset X;
    a"\/"b in Y by A1,A2,WAYBEL_0:40;
    hence x \/ y in Y by YELLOW_1:17;
  end;
  assume
A3: for x,y being set st x in Y & y in Y holds x \/ y in Y;
  now
    let a,b be Element of BoolePoset X;
    assume a in Y & b in Y;
    then a \/ b in Y by A3;
    hence a"\/"b in Y by YELLOW_1:17;
  end;
  hence thesis by WAYBEL_0:40;
end;
