reserve i for object, I for set,
  f for Function,
  x, x1, x2, y, A, B, X, Y, Z for ManySortedSet of I;

theorem     :: SYSREL:17
  X (/\) Y = EmptyMS I implies [|X,Y|] (/\) [|Y,X|] = EmptyMS I
proof
  assume
A1: X (/\) Y = EmptyMS I;
  now
    let i be object;
    assume
A2: i in I;
    then X.i /\ Y.i = (X (/\) Y).i by PBOOLE:def 5
      .= {} by A1,PBOOLE:5;
    then X.i misses Y.i by XBOOLE_0:def 7;
    then
A3: [:X.i,Y.i:] misses [:Y.i,X.i:] by ZFMISC_1:104;
    thus ([|X,Y|] (/\) [|Y,X|]).i = [|X,Y|].i /\ [|Y,X|].i by A2,PBOOLE:def 5
      .= [:X.i,Y.i:] /\ [|Y,X|].i by A2,PBOOLE:def 16
      .= [:X.i,Y.i:] /\ [:Y.i,X.i:] by A2,PBOOLE:def 16
      .= {} by A3,XBOOLE_0:def 7
      .= EmptyMS I.i by PBOOLE:5;
  end;
  hence thesis;
end;
