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

theorem :: ZFMISC_1:86
  bool (A (\) B) (\/) bool (B (\) A) c= bool (A (\+\) B)
proof
  let i be object;
  assume
A1: i in I;
  then
A2: bool (A (\+\) B).i = bool (A.i \+\ B.i) by Lm5;
  (bool (A (\) B) (\/) bool (B (\) A)).i
     = (bool (A (\) B)).i \/ (bool (B (\) A) ).i
  by A1,PBOOLE:def 4
    .= (bool (A.i \ B.i)) \/ (bool (B (\) A)).i by A1,Lm4
    .= (bool (A.i \ B.i)) \/ (bool (B.i \ A.i)) by A1,Lm4;
  hence thesis by A2,ZFMISC_1:73;
end;
