theorem
  bool (A /\ B) = bool A /\ bool B
proof
  now
    let x;
     reconsider xx=x as set by TARSKI:1;
    A /\ B c= A & A /\ B c= B by XBOOLE_1:17;
    then xx c= A & xx c= B iff xx c= A /\ B by XBOOLE_1:19;
    then x in bool A & x in bool B iff x in bool (A /\ B) by Def1;
    hence x in bool (A /\ B) iff x in bool A /\ bool B by XBOOLE_0:def 4;
  end;
  hence thesis by TARSKI:2;
end;
