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 Th49:
  for i being Nat
  for p being FinSequence of NAT st not i in dom p
  holds primrec(e1,e2,i,p) = {}
proof let i be Nat;
  set f1 = e1, f2 = e2;
  let p be FinSequence of NAT;
  consider F being sequence of  HFuncs NAT such that
A1: primrec(f1,f2,i,p) = F.(p/.i) and
  i in dom p & Del(p,i) in dom f1 implies F.0 = p+*(i,0) .--> (f1.Del(p,i)
  ) and
A2: not i in dom p or not Del(p,i) in dom f1 implies F.0 = {} and
A3: for m being Nat holds Q[m, F.m qua Element of HFuncs NAT,
  F.(m+1)qua Element of HFuncs NAT, p, i, f2] by Def10;
  defpred p[Nat] means F.$1 = {};
  assume
A4: not i in dom p;
  then
A5: for m be Nat st p[m] holds p[m+1] by A3;
A6: p[0] by A4,A2;
  for m being Nat holds p[m] from NAT_1:sch 2(A6, A5);
  hence thesis by A1;
end;
