theorem Th22:
  for f being Function holds rng("f) c= bool dom f
proof
  let f be Function;
  let x be object;
   reconsider xx=x as set by TARSKI:1;
  assume x in rng("f);
  then consider y being object such that
A1: y in dom("f) & x = "f.y by FUNCT_1:def 3;
   reconsider y as set by TARSKI:1;
  x = f"y by A1,Th21;
  then xx c= dom f by RELAT_1:132;
  hence thesis;
end;
