reserve I for set;

theorem Th14:
  for f being ManySortedFunction of I holds f""(rngs f) = doms f
proof
  let f be ManySortedFunction of I;
  now
    let i be object;
    assume
A1: i in I;
    hence (f""(rngs f)).i = (f.i)"((rngs f).i) by Def1
      .= (f.i)"(rng (f.i)) by A1,MSSUBFAM:13
      .= dom (f.i) by RELAT_1:134
      .= (doms f).i by A1,MSSUBFAM:14;
  end;
  hence thesis;
end;
