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;
reserve P for primitive-recursively_closed non empty Subset of HFuncs NAT;

theorem Th70:
  {} in P
proof
  reconsider F = {} as with_the_same_arity Element of (HFuncs NAT)* by
FINSEQ_1:49;
  set f = 0 const 0;
A1: rng {} c= P;
A2: arity f = 0;
A3: P is composition_closed by Def14;
  f in P by Th68;
  then f*<:F:> in P by A2,A1,A3,CARD_1:27;
  hence thesis by FUNCT_6:40;
end;
