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

theorem :: ZFMISC_1:83
  bool (A (/\) B) = bool A (/\) bool B
proof
  now
    let i be object;
    assume
A1: i in I;
    hence bool (A (/\) B).i = bool (A.i /\ B.i) by Lm3
      .= (bool (A.i)) /\ (bool (B.i)) by ZFMISC_1:71
      .= (bool (A.i)) /\ (bool B.i) by A1,Def1
      .= (bool A).i /\ (bool B.i) by A1,Def1
      .= (bool A (/\) bool B).i by A1,PBOOLE:def 5;
  end;
  hence thesis;
end;
