
theorem Th27:
  for N being non empty reflexive RelStr, X, Y being Subset of N
  holds (X^0) \/ (Y^0) c= (X \/ Y)^0
proof
  let N be non empty reflexive RelStr, X, Y be Subset of N;
  let a be object;
  assume
A1: a in (X^0) \/ (Y^0);
  then reconsider b = a as Element of N;
  now
    let D be non empty directed Subset of N such that
A2: b <= sup D;
    now
      per cases by A1,XBOOLE_0:def 3;
      suppose
        a in X^0;
        then ex x being Element of N st a = x & for D being
        non empty directed Subset of N st x <= sup D holds X meets D;
        then X meets D by A2;
        then X /\ D <> {};
        then X /\ D \/ Y /\ D <> {};
        hence (X \/ Y) /\ D <> {} by XBOOLE_1:23;
      end;
      suppose
        a in Y^0;
        then ex y being Element of N st a = y & for D being
        non empty directed Subset of N st y <= sup D holds Y meets D;
        then Y meets D by A2;
        then Y /\ D <> {};
        then X /\ D \/ Y /\ D <> {};
        hence (X \/ Y) /\ D <> {} by XBOOLE_1:23;
      end;
    end;
    hence (X \/ Y) meets D;
  end;
  hence thesis;
end;
