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

theorem     :: ZFMISC_1:75
  X = EmptyMS I or X = {x} or X = {y} or X = {x,y}
    implies X (\) {x,y} = EmptyMS I
proof
  assume
A1: X = EmptyMS I or X = {x} or X = {y} or X = {x,y};
  now
    let i be object such that
A2: i in I;
    per cases by A1;
    suppose X = EmptyMS I;
      then
A3:   X.i = {} by PBOOLE:5;
      thus (X (\) {x,y}).i = X.i \ {x,y}.i by A2,PBOOLE:def 6
        .= EmptyMS I.i by A3,PBOOLE:5;
    end;
    suppose X = {x};
      then
A4:   X.i = {x.i} by A2,Def1;
      thus (X (\) {x,y}).i = X.i \ {x,y}.i by A2,PBOOLE:def 6
        .= X.i \ {x.i,y.i} by A2,Def2
        .= {} by A4,ZFMISC_1:66
        .= EmptyMS I.i by PBOOLE:5;
    end;
    suppose X = {y};
      then
A5:   X.i = {y.i} by A2,Def1;
      thus (X (\) {x,y}).i = X.i \ {x,y}.i by A2,PBOOLE:def 6
        .= X.i \ {x.i,y.i} by A2,Def2
        .= {} by A5,ZFMISC_1:66
        .= EmptyMS I.i by PBOOLE:5;
    end;
    suppose X = {x,y};
      then
A6:   X.i = {x.i,y.i} by A2,Def2;
      thus (X (\) {x,y}).i = X.i \ {x,y}.i by A2,PBOOLE:def 6
        .= X.i \ {x.i,y.i} by A2,Def2
        .= {} by A6,ZFMISC_1:66
        .= EmptyMS I.i by PBOOLE:5;
    end;
  end;
  hence thesis;
end;
