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 Th79:
  f1 is len-total & arity f1 = 0 implies primrec(f1,f2,1).<*0*> = f1.{}
proof
  assume that
A1: f1 is len-total and
A2: arity f1 = 0;
  reconsider p = <*0*> as Element of (arity f1 +1)-tuples_on NAT
  by A2,FINSEQ_2:131;
  len p = 1 by FINSEQ_1:39;
  then
A3: 1 in dom p by FINSEQ_3:25;
  p+*(1,0) = p by FUNCT_7:95;
  hence primrec(f1,f2,1).<*0*> = f1.Del(p,1) by A1,A3,Th59
    .= f1.{} by WSIERP_1:19;
end;
