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) c= { {} } \/ (bool A \ bool B)
proof
  let x;
   reconsider xx=x as set by TARSKI:1;
  assume x in bool (A \ B);
  then
A1: xx c= A \ B by Def1;
  then xx misses B by XBOOLE_1:63,79;
  then A \ B c= A & xx /\ B = {} by XBOOLE_1:36;
  then x = {} or xx c= A & not xx c= B by A1,XBOOLE_1:28;
  then x in { {} } or x in bool A & not x in bool B by Def1,TARSKI:def 1;
  then x in { {} } or x in bool A \ bool B by XBOOLE_0:def 5;
  hence thesis by XBOOLE_0:def 3;
end;
