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 Th71:
  for f being Element of P, F being with_the_same_arity FinSequence of P
   st arity f = len F holds f*<:F:> in P
proof
  let f be Element of P, F being with_the_same_arity FinSequence of P;
  assume
A1: arity f = len F;
A2: rng F c= P by FINSEQ_1:def 4;
  per cases;
  suppose
    f is empty;
    then f*<:F:> = {};
    hence thesis by Th70;
  end;
  suppose
    f is non empty;
    then reconsider f9 = f as non empty Element of HFuncs NAT;
A3: P* c= (HFuncs NAT)* by FINSEQ_1:62;
    F in P* by FINSEQ_1:def 11;
    then reconsider F9 = F as with_the_same_arity Element of (HFuncs NAT)*
    by A3;
    P is composition_closed by Def14;
    then f9*<:F9:> in P by A1,A2;
    hence thesis;
  end;
end;
