reserve p,q,x,x1,x2,y,y1,y2,z,z1,z2 for set;
reserve A,B,V,X,X1,X2,Y,Y1,Y2,Z for set;
reserve C,C1,C2,D,D1,D2 for non empty set;

theorem
  for f being Function holds (.:f).:A c= bool rng f
proof
  let f be Function;
  (.:f).:A c= rng(.:f) & rng(.:f) c= bool rng f by Th9,RELAT_1:111;
  hence thesis;
end;
