theorem
  (A?) ^^ (A |^ k) = A |^ (k, k + 1)
proof
  (A |^ (0, 1)) ^^ (A |^ k) = (A |^ (0, 1)) ^^ (A |^ (k, k)) by Th22
    .= A |^ (0 + k, 1 + k) by Th37;
  hence thesis by Th79;
end;
