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 Th62:
  f1 is len-total & f2 is len-total & arity f1 +2 = arity f2 &
  1 <= i & i <= 1+arity f1 implies
  primrec(f1,f2,i).(p+*(i,m+1)) =
   f2.((p+*(i,m))^<*primrec(f1,f2,i).(p+*(i,m))*>)
proof
  assume that
A1: f1 is len-total and
A2: f2 is len-total and
A3: arity f1 +2 = arity f2 and
A4: 1 <= i and
A5: i <= 1+arity f1;
  len p = arity f1 +1 by CARD_1:def 7;
  then
A6: i in dom p by A4,A5,FINSEQ_3:25;
  (p+*(i,m+1)) in (arity f1 +1)-tuples_on NAT;
  then (p+*(i,m+1)) in dom primrec(f1,f2,i) by A1,A2,A3,A4,A5,Th56;
  hence thesis by A6,Lm6;
end;
