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 (B \ A) c= bool (A \+\ B)
proof
  let x;
   reconsider xx=x as set by TARSKI:1;
  assume x in bool (A \ B) \/ bool (B \ A);
  then x in bool (A \ B) or x in bool (B \ A) by XBOOLE_0:def 3;
  then
A1: xx c= A \ B or xx c= B \ A by Def1;
  xx c= (A \ B) \/ (B \ A)
  proof
    let z;
    assume z in xx;
    then z in A \ B or z in B \ A by A1;
    hence thesis by XBOOLE_0:def 3;
  end;
  hence thesis by Def1;
end;
