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_1:20
  I is non empty & {x} (\) {y} = {x} implies x <> y
proof
  assume that
A1: I is non empty and
A2: {x} (\) {y} = {x};
  now
    consider i being object such that
A3: i in I by A1,XBOOLE_0:def 1;
    take i;
    {x.i} \ {y.i} = {x}.i \ {y.i} by A3,Def1
      .= {x}.i \ {y}.i by A3,Def1
      .= ({x} (\) {y}).i by A3,PBOOLE:def 6
      .= {x.i} by A2,A3,Def1;
    hence x.i <> y.i by ZFMISC_1:14;
  end;
  hence thesis;
end;
