As in Construction 2.6, we cover $\mathfrak{X}^{\prime}$ by étale maps $\psi:\mathfrak{U}\rightarrow\mathfrak{X}(\boldsymbol{n},\boldsymbol{a},m)=\mathfrak{X}(\boldsymbol{n},\boldsymbol{a})\times\mathfrak{X}(m)$ and for each $\mathfrak{U}$ and $\Delta\in\mathfrak{D}$ with $\Delta\subseteq\mathfrak{U}^{\textup{an}}$ we obtain the affine formal scheme $\mathfrak{U}_{\Delta}$. We write $\psi^{\prime}:\mathfrak{U}^{\prime\prime}_{\Delta}\rightarrow\mathfrak{U}_{\Delta}$ for the base change with respect to $\psi$ and obtain a cover of $\mathfrak{X}^{\prime\prime}$. On $\Delta\in\mathfrak{D}$, $h$ is given by $\boldsymbol{m}\boldsymbol{x}+v(\alpha)$ with $\boldsymbol{m}\in\mathbb{Z}^{\boldsymbol{n}+\boldsymbol{1}}$, $\alpha\in K^{\times}$. We define $D$ locally on $\mathfrak{U}^{\prime\prime}_{\Delta^{\prime}}$ by $\psi^{\prime\ast}(\alpha\cdot\boldsymbol{x}^{\boldsymbol{m}})$. Then $D$ is indeed a Cartier Divisor on $\mathfrak{X}^{\prime\prime}$ as for $\mathfrak{U}_{1},\mathfrak{U}_{2},\Delta_{1},\Delta_{2}$ as above and $\mathfrak{U}:=\mathfrak{U}_{1}\cap\mathfrak{U}_{2}$ we have $\alpha_{1}\cdot\boldsymbol{x}^{\boldsymbol{m}_{1}}/\alpha_{2}\cdot\boldsymbol{x}^{\boldsymbol{m}_{2}}\in\mathcal{O}(\mathfrak{U}_{\Delta_{1}\cap\Delta_{2}})^{\times}$ since $\boldsymbol{m}_{1}\boldsymbol{x}+v(\alpha_{1})=\boldsymbol{m}_{2}\boldsymbol{x}+v(\alpha_{2})$ on $\Delta_{1}\cap\Delta_{2}$. Hence

$\displaystyle\psi_{1}^{\prime\ast}(\alpha_{1}\cdot\boldsymbol{x}^{\boldsymbol{m}_{1}})/\psi_{2}^{\prime\ast}(\alpha_{2}\cdot\boldsymbol{x}^{\boldsymbol{m}_{2}})\Big|_{\mathfrak{U}^{\prime\prime}_{\Delta_{1}\cap\Delta_{2}}}=\psi^{\prime\ast}(\alpha_{1}\cdot\boldsymbol{x}^{\boldsymbol{m}_{1}}/\alpha_{2}\cdot\boldsymbol{x}^{\boldsymbol{m}_{2}})\in\mathcal{O}(\mathfrak{U}^{\prime\prime}_{\Delta_{1}\cap\Delta_{2}})^{\times}$

and therefore $\psi_{1}^{\prime\ast}(\alpha_{1}\boldsymbol{x}^{\boldsymbol{m}_{1}})/\psi_{2}^{\prime\ast}(\alpha_{2}\boldsymbol{x}^{\boldsymbol{m}_{2}})\in\mathcal{O}(\mathfrak{U}^{\prime\prime}_{1,\Delta_{1}}\cap\mathfrak{U}^{\prime\prime}_{2,\Delta_{2}})^{\times}$. Furthermore $D$ is trivial on the generic fibre, as $\alpha\cdot\boldsymbol{x}^{\boldsymbol{m}}\in\mathcal{O}(\mathfrak{X}(\boldsymbol{n},\boldsymbol{a})^{\textup{an}})^{\times}$. ∎