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 Th72:
  for f1,f2 being Element of P st arity f1+2 = arity f2
  for i being Nat st
   1 <= i & i <= arity f1+1 holds primrec(f1,f2,i) in P
proof
  let f1,f2 be Element of P;
  assume
A1: arity f1+2 = arity f2;
  let i be Nat;
  assume that
A2: 1 <= i and
A3: i <= arity f1+1;
A4: P is primitive-recursion_closed by Def14;
  per cases;
  suppose
    f1 is empty;
    then primrec(f1,f2,i) is empty by Th54;
    hence thesis by Th70;
  end;
  suppose
    f1 is non empty;
    then primrec(f1,f2,i) is_primitive-recursively_expressed_by f1,f2,i
    by A1,A2,A3,Th17,Th63;
    hence thesis by A4;
  end;
end;
