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 Th66:
  for g being Element of HFuncs NAT st e1 = {} &
  g is_primitive-recursively_expressed_by e1, e2, i holds g = {}
proof
  set f1 = e1, f2 = e2;
  let g be Element of HFuncs(NAT);
  assume
A1: f1 = {};
  assume g is_primitive-recursively_expressed_by f1, f2, i;
  then consider n being Element of NAT such that
A2: dom g c= n-tuples_on NAT and
  i >= 1 and
  i <= n and
  (arity f1)+1 = n and
  n+1 = arity f2 and
A3: for p being FinSequence of NAT st len p = n holds (p+*(i,0) in dom g
iff Del(p,i) in dom f1) & (p+*(i,0) in dom g implies g.(p+*(i,0)) = f1.Del(p,i)
) & for n being Nat holds (p+*(i,n+1) in dom g iff p+*(i,n) in dom g
& (p+*(i,n))^<*g.(p+*(i,n))*> in dom f2) & (p+*(i,n+1) in dom g implies g.(p+*(
  i,n+1)) = f2.((p+*(i,n))^<*g.(p+*(i,n))*>));
A4: now
    let y be Element of n-tuples_on NAT;
    defpred p[Nat] means not y+*(i,$1) in dom g;
A5: len y = n by CARD_1:def 7;
    then
A6: for k being Nat st p[k] holds p[k+1] by A3;
A7: p[0] by A1,A3,A5;
    thus for k being Nat holds p[k] from NAT_1:sch 2(A7, A6);
  end;
  assume g <> {};
  then consider x being object such that
A8: x in dom g by XBOOLE_0:def 1;
  reconsider x as Element of n-tuples_on NAT by A2,A8;
  set xi = x.i;
  x = x+*(i,xi) by FUNCT_7:35;
  hence contradiction by A4,A8;
end;
