reserve I for set;

theorem Th13:
  for f being ManySortedFunction of I holds f.:.:(doms f) = rngs f
proof
  let f be ManySortedFunction of I;
  now
    let i be object;
    assume
A1: i in I;
    hence (f.:.:(doms f)).i = (f.i).:((doms f).i) by PBOOLE:def 20
      .= (f.i).:(dom (f.i)) by A1,MSSUBFAM:14
      .= rng (f.i) by RELAT_1:113
      .= (rngs f).i by A1,MSSUBFAM:13;
  end;
  hence thesis;
end;
