reserve u,v,x,x1,x2,y,y1,y2,z,p,a for object,
        A,B,X,X1,X2,X3,X4,Y,Y1,Y2,Z,N,M for set;

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;
