reserve i, j, k, c, m, n for Nat,
  a, x, y, z, X, Y for set,
  D, E for non empty set,
  R for Relation,
  f, g for Function,
  p, q for FinSequence;

theorem Th27:
  for f being to-naturals homogeneous NAT*-defined Function holds
  f is Element of HFuncs NAT
proof
  let f be to-naturals homogeneous NAT*-defined Function;
A1: rng f c= NAT by VALUED_0:def 6;
  dom f c= NAT*;
  then f is PartFunc of NAT*,NAT by A1,RELSET_1:4;
  then reconsider f9=f as Element of PFuncs(NAT*,NAT) by PARTFUN1:45;
  f9 in {f1 where f1 is Element of PFuncs(NAT*,NAT): f1 is homogeneous};
  hence thesis;
end;
