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_1:55
  I is non empty & {x,y} (/\) X = EmptyMS I implies not x in X & not y in X
proof
  assume that
A1: I is non empty and
A2: {x,y} (/\) X = EmptyMS I;
A3: now
    let i;
    assume
A4: i in I;
    hence {x.i,y.i} /\ X.i = {x,y}.i /\ X.i by Def2
      .= ({x,y} (/\) X).i by A4,PBOOLE:def 5
      .= {} by A2,PBOOLE:5;
  end;
  thus not x in X
  proof
    assume
A5: x in X;
    now
      consider i being object such that
A6:   i in I by A1,XBOOLE_0:def 1;
      take i;
      {x.i,y.i} /\ X.i = {} by A3,A6;
      then {x.i,y.i} misses X.i by XBOOLE_0:def 7;
      hence i in I & not x.i in X.i by A6,ZFMISC_1:49;
    end;
    hence contradiction by A5;
  end;
  assume
A7: y in X;
  now
    consider i being object such that
A8: i in I by A1,XBOOLE_0:def 1;
    take i;
    {x.i,y.i} /\ X.i = {} by A3,A8;
    then {x.i,y.i} misses X.i by XBOOLE_0:def 7;
    hence i in I & not y.i in X.i by A8,ZFMISC_1:49;
  end;
  hence contradiction by A7;
end;
