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
  i in dom p implies (p+*(i,m+1) in dom primrec(f1,f2,i) iff
  p+*(i,m) in dom primrec(f1,f2,i) &
  (p+*(i,m))^<*primrec(f1,f2,i).(p+*(i,m))*> in dom f2) by Lm6;
