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_1:65
  I is non empty & X (\) {x} = X implies not x in X
proof
  assume that
A1: I is non empty and
A2: X (\) {x} = X and
A3: x in X;
A4: now
    let i;
    assume
A5: i in I;
    hence X.i \ {x.i} = X.i \ {x}.i by Def1
      .= X.i by A2,A5,PBOOLE:def 6;
  end;
  now
    consider i being object such that
A6: i in I by A1,XBOOLE_0:def 1;
    take i;
    thus i in I by A6;
    X.i \ {x.i} = X.i by A4,A6;
    hence not x.i in X.i by ZFMISC_1:57;
  end;
  hence contradiction by A3;
end;
