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 Th52:
  for p being FinSequence of NAT st e1 is empty
   holds primrec(e1,e2,i,p) is empty
proof
  set f1 = e1, f2 = e2;
  let p be FinSequence of NAT;
  consider F be 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 (i in dom p & p+*(i,m) in dom (F.m)
& (p+*(i,m))^<*(F.m).(p+*(i,m))*> in dom f2 implies F.(m+1) = (F.m)+*(p+*(i,m+1
).--> f2.((p+*(i,m))^<*(F.m).(p+*(i,m))*>))) & (not i in dom p or not p+*(i,m)
in dom (F.m) or not (p+*(i,m))^<*(F.m).(p+*(i,m))*> in dom f2 implies F.(m+1) =
  F.m) by Def10;
  defpred p[Nat] means F.$1 = {};
A4: for k be Nat st p[k] holds p[k+1] by A3,RELAT_1:38;
  assume f1 is empty;
  then
A5: p[0] by A2;
  for k being Nat holds p[k] from NAT_1:sch 2(A5,A4);
  hence thesis by A1;
end;
