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_1:64 b
  I is non empty & y in X (\) {x} implies y <> x
proof
  assume that
A1: I is non empty and
A2: y in X (\) {x};
  consider i being object such that
A3: i in I by A1,XBOOLE_0:def 1;
  now
    take i;
    y.i in (X (\) {x}).i by A2,A3;
    then y.i in X.i \ {x}.i by A3,PBOOLE:def 6;
    then y.i in X.i \ {x.i} by A3,Def1;
    hence y.i <> x.i by ZFMISC_1:56;
  end;
  hence thesis;
end;
