there exists a non-empty open set $U_\eta$ of the component $X_\eta$ of the special fiber, such that $\alpha \otimes F_{U_\eta}(\sqrt[p]{f}, 
 \sqrt[p^2]{u_1}, \cdots , \sqrt[p^2]{u_{d}}, \sqrt[p]{u_{d+1}}) 
 = 0$ for all $i$.

Let ${{{\mathscr{P}}}}$ be the finite set of closed points of ${{{\mathscr{X}}}}$ which are not in $U_\eta$ for any $\eta$. Let $A$ be the semi-local ring at the points of ${{\mathscr{P}}}$. For $P \in {{\mathscr{P}}}$, let $A_P$ be the local ring at $P$. Since the ramifications of $\alpha_i$ for all $i$ are in normal crossings, for each $P  \in {{{\mathscr{P}}}}$, the maximal ideal $m_p$ at $P$ is $(\pi_P, \delta_P)$ for some $\pi_P$ and $\delta_P$ such that $\alpha_i$ is unramified on $A_P$ except possibly at $(\pi_P)$ and $(\delta_P)$. Since the residue field $\kappa(P)$ at $P$ is a finite extension of $\kappa$ and $[\kappa : \kappa^p] = p^d$, $[\kappa(P) :
\kappa(P)^p] = p^d$ ([@bourbaki:algebra; @ll A.V.135, Corollary 3]). Let $v_{P, 1}, \cdots , v_{P, d} \in A_P^*$ be lifts of a $p$-basis of $\kappa(P)$. By the Chinese remainder theorem, choose $h_1, \cdots , h_d \in A^*$ such that $h_i = v_{P, i}$ modulo the maximal ideal at $P$ for all $P
\in {{\mathscr{P}}}$ and $1\leq i
\leq d$. By (\[lemma:choice\_of\_f\_g\]), there exist $g_1, 