 that $G$ and $\mathcal{R}$ form a regular sequence in ${\overline{{\mathbb F}}_{\hskip-0.0mm q}}(A_r {\ifm {,\ldots,} {$,\ldots,$}}A_0)[A_0,A_0]$, and Lemma \[lemma: regular sequences 0\] implies that $G$ and $\mathcal{R}$ form a regular sequence in ${\overline{{\mathbb F}}_{\hskip-0.0mm q}}[A_r {\ifm {,\ldots,} {$,\ldots,$}}A_0]$. 