reserve x, y for object, I for set,
  A, B, X, Y for ManySortedSet of I;

theorem :: ZFMISC_1:84
  bool (A (\) B) c= (I --> {{}}) (\/) (bool A (\) bool B)
proof
  let i be object;
  assume
A1: i in I;
  then
A2: (bool (A (\) B)).i = bool (A.i \ B.i) by Lm4;
  ((I --> {{}}) (\/) (bool A (\) bool B)).i
      = (I --> {{}}).i \/ (bool A (\) bool B).i by A1,PBOOLE:def 4
    .= {{}} \/ (bool A (\) bool B).i by A1,FUNCOP_1:7
    .= {{}} \/ ((bool A).i \ (bool B).i) by A1,PBOOLE:def 6
    .= {{}} \/ ((bool (A.i)) \ (bool B).i) by A1,Def1
    .= {{}} \/ (bool (A.i) \ bool (B.i)) by A1,Def1;
  hence thesis by A2,ZFMISC_1:72;
end;
