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_1:30
  bool { x } = { EmptyMS I, {x} }
proof
  now
    let i be object;
    assume
A1: i in I;
    hence (bool {x}).i = bool ({x}.i) by MBOOLEAN:def 1
      .= bool ({x.i}) by A1,Def1
      .= {{}, {x.i}} by ZFMISC_1:24
      .= {EmptyMS I.i, {x.i}} by PBOOLE:5
      .= {EmptyMS I.i, {x}.i} by A1,Def1
      .= { EmptyMS I, {x} }.i by A1,Def2;
  end;
  hence thesis;
end;
