is finite flat being the rigidification of a map of affine schemes $T_L\rightarrow T_L/\mathcal{S}_n$ which is finite flat. Consequently the inverse image of $\mathrm{Sp}\,A$ in $T_L^{\rm rig}$ is an admissible affinoid open subspace $\mathrm{Sp}\,B$ with $B$ an affinoid algebra which is finite flat over $A$. As $B$ is a finite $A$-algebra, we have an isomorphism $C\widehat{\otimes}_AB\simeq C\otimes_AB$. It follows, by definition of the fiber product of rigid analytic spaces, that the rigid analytic spaces of the form $\mathrm{Sp}(C\otimes_AB)$ form an admissible open covering of $\widetilde{\mathfrak{X}}_{\overline r}^{\square,{\bf k}\rm -cr}=\mathfrak{X}_{\overline{r}}^{\square,{\rm {\bf k}-cr}}\times_{T_L^{\rm rig}/\mathcal{S}_n}T_L^{\rm rig}$. It is sufficient to prove that rings $C\otimes_AB$ as above are reduced. From Lemma \[red\] below it is sufficient to prove that $C\otimes_AB$ is a finite flat generically étale $C$-algebra. As $B$ is finite flat over $A$, the $C$-algebra $C\otimes_AB$ is clearly finite flat. It is sufficient to prove that it is a generically étale $C$-algebra. As $B$ is generically étale over $A$, it is sufficient to prove that the map $\mathrm{Spec}\,C\rightarrow\mathrm{Spec}\,A$ is dominant. It is thus sufficient to prove that the map of rigid analytic spaces $\mathfrak{X}_{\overline{r}}^{\square,{\rm {\bf k}-cr}}\rightarrow T_L^{\rm rig}/\mathcal{S}_n$ is open. This follows from the fact that it has, locally on $\mathfrak{X}_{\overline{r}}^{\square,{\rm {\bf k}-cr}}$, a 