$|S_i|={l(b_i)}$ for which $S_0 \cup S_1$ is an interval of the form $\{1,\ldots,k\}$ for some $k$, which is then at most ${l(b_0)}+{l(b_1)}$. Note that there are only finitely many such pairs $(S_0,S_1)$. For each such pair let $(\pi_0,\pi_1)$ be a pair of elements of ${\mathrm{Inc}({{\mathbb N}})}$ such that $\pi_i$ maps $\{1,\ldots,{l(b_i)}\}$ onto $S_i$; it is irrelevant how $\pi$ acts on the rest of ${{\mathbb N}}$. Then we have $${\mathrm{Inc}({{\mathbb N}})}b_0 \times {\mathrm{Inc}({{\mathbb N}})}b_1 = \bigcup_{(S_0,S_1)} {\mathrm{Inc}({{\mathbb N}})}(\pi_0 b_0,\pi_1
b_1),$$ where the union is over all pairs $(S_0,S_1)$ as above.

The $42$ polynomials of $B$ were constructed by computing a Gröbner basis for $J_9({{\mathbb Q}})$ with [Singular]{} [@GPS05] and retaining only those polynomials $p$ for which the set of indices occurring in their variables form an interval of the form $\{1,\ldots,k\}$ with $k \leq 9$. All elements of $B$ are monic and have integral coefficients (in fact, equal to $\pm 1$ except for the $3 \times 3$-minor with largest index $3$, which has a coefficient $2$). By the equivariant Buchberger criterion and the proof of Lemma \[lm:FiniteS\], we need only ${\mathrm{Inc}({{\mathbb N}})}$-reduce modulo $B$ all S-polynomials of pairs $(\pi_0b_0,\pi_1b_1)$ with $b_0,b_1 \in B$ and $\pi_i:\{1,\ldots,{l(b_i)}\} \to {{\mathbb N}}$ increasing and such 