Proposition. (Generalized Associative Law) Let $k$ and $n$ be two natural numbers with $k < n$, and let $x_{k+1}, \ldots, x_n$ be a sequence of natural numbers. In addition, let $r \geq 1$ be a natural number, and let $n_0, n_1, \ldots, n_r$ be a sequence of natural numbers such that