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 Th59:
  i in dom p & f1 is len-total implies
  primrec(f1,f2,i).(p+*(i,0)) = f1.Del(p,i)
proof
  assume that
A1: i in dom p and
A2: f1 is len-total;
A3: Del(p,i) is FinSequence of NAT by FINSEQ_3:105;
  len p = arity f1+1 by CARD_1:def 7;
  then len Del(p,i) = arity f1 by A1,FINSEQ_3:109;
  then
A4: Del(p,i) is Element of (arity f1)-tuples_on NAT by A3,FINSEQ_2:92;
  dom f1 = (arity f1)-tuples_on NAT by A2,Th22;
  then p+*(i,0) in dom primrec(f1,f2,i) by A1,A4,Lm6;
  hence thesis by A1,Lm6;
end;
