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 Th16:
  for g being len-total to-naturals NAT*-defined Function holds
  g is quasi_total PartFunc of NAT*, NAT
proof
  let g be len-total to-naturals NAT*-defined Function;
A1: rng g c= NAT by VALUED_0:def 6;
  dom g c= NAT*;
  then reconsider g9 = g as PartFunc of NAT*, NAT by A1,RELSET_1:4;
  for x,y being FinSequence of NAT st len x = len y & x in dom g holds y
  in dom g by Def2;
  then g9 is quasi_total;
  hence thesis;
end;
