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;
reserve f1, f2 for non empty homogeneous to-naturals NAT*-defined Function,
  e1, e2 for homogeneous to-naturals NAT*-defined Function,
  p for Element of (arity f1+1)-tuples_on NAT;

theorem Th65:
  HFuncs NAT is primitive-recursively_closed
proof
  set X = HFuncs NAT;
  thus 0 const 0 in X & 1 succ 1 in X & for n,i being Nat st 1<=i &
  i <= n holds n proj i in X by Th30,Th32,Th34;
  thus for f being Element of HFuncs NAT for F being with_the_same_arity
FinSequence of HFuncs NAT st f in X & arity f = len F & rng F c= X holds f*<:F
  :> in X by Th41;
  let g be Element of HFuncs NAT;
  thus thesis;
end;
