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 a
  y in X (\) {x} implies y in X
proof
  assume
A1: y in X (\) {x};
  let i;
  assume
A2: i in I;
  then y.i in (X (\) {x}).i by A1;
  then y.i in X.i \ {x}.i by A2,PBOOLE:def 6;
  then y.i in X.i \ {x.i} by A2,Def1;
  hence thesis by ZFMISC_1:56;
end;
