reserve E, x, y, X for set;
reserve A, B, C for Subset of E^omega;
reserve a, b for Element of E^omega;
reserve i, k, l, kl, m, n, mn for Nat;

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;
