Lemma 2

Let $A$ be a graded commutative ring such that $A = A_0[A_1]$ and let $S = \bigoplus_{i \geq 0} S_i$, be a graded commutative $A$-algebra with positive degrees, which is a finitely generated $A$-module. Then there exists an integer $n_0 \geq 0$ such that:

(i) For $n \geq n_0$ and $k \geq 0$, $S_{n+k} = S_k \cdot S_n$. (ii) For $d \geq n_0$, $S^{(d)} = S_0[S_d]$.

By Lemma 1 there exists an integer $n_0 \geq 0$ such that, for $n \geq n_0$ and $k \geq 0$, $S_{n+k} = A_k S_n$, whence a fortiori $S_{n+k} = S_k S_n$, which establishes (i). Then, for $d \geq n_0$ and $m > 0$, $S_{m d} = (S_d)^m$ as follows by induction on $m$ applying (i); this establishes (ii).