We show that $s^{\sigma}_{i}$ is an equation for $D_{i}$ on $\mathfrak{X}_{\sigma}$; the proof is analogous for $s^{\sigma}_{j}$.

On $Z$, ${W_{i}^{\sigma}}_{|Z}=\textrm{div}(h)$ and $s^{\sigma}_{i}$ is a non-zero global section, which means that

	$\displaystyle{\mathscr{L}^{\sigma}_{i}}_{|Z}(Z)=\mathcal{O}_{Z}({W^{\sigma}_{i}})(Z)=\{f\in\mathcal{K}(Z)\,|\,\textrm{div}(f)+\textrm{div}(h)\geqslant 0\}$	$\displaystyle\xrightarrow{\simeq}\mathcal{O}_{Z}(Z)=k$
	$\displaystyle f$	$\displaystyle\mapsto fh$
	$\displaystyle s^{\sigma}_{i}$	$\displaystyle\mapsto s^{\sigma}_{i}h=\lambda\in k^{\times}.$

Let $\mathcal{U}$ be an open cover of $\mathscr{X}\setminus\big(\cup_{i^{\prime}\notin J\cup L_{\sigma}}D_{i^{\prime}}\big)$ such that ${D_{i}}_{|U}=\textrm{div}(g_{U})$ for any $U\in\mathcal{U}$; this is possible as $D_{i}$ is a Cartier divisor. On $U$, ${W^{\sigma}_{i}}_{|U}=-{D_{i}}_{|U}=\textrm{div}(g_{U}^{-1})$ and

	$\displaystyle\mathscr{L}^{\sigma}_{i}(\mathfrak{X}_{\sigma}\cap U)$	$\displaystyle\xrightarrow{\simeq}\mathcal{O}_{\mathfrak{X}_{\sigma}}(\mathfrak{X}_{\sigma}\cap U)$
	$\displaystyle f$	$\displaystyle\mapsto fg_{U}^{-1}$
	$\displaystyle s^{\sigma}_{i}$	$\displaystyle\mapsto s^{\sigma}_{i}g_{U}^{-1}\in\mathcal{O}_{\mathfrak{X}_{\sigma}}^{\times}(\mathfrak{X}_{\sigma}\cap U),$

where $s^{\sigma}_{i}g_{U}^{-1}$ is a regular invertible function on $\mathfrak{X}_{\sigma}\cap U$, as its reduction to $Z$ is invertible. Finally, the section $s^{\sigma}_{i}$ is defined globally on $\widehat{\mathscr{X}_{/Z}}$ and on each open $\mathfrak{X}_{\sigma}\cap U$ gives a local equation of the divisor $D_{i}$, hence it is a equation for $D_{i}$ on $\mathfrak{X}_{\sigma}$. ∎