\cdots \times _{Y_0} X_0}_{s}$$ of $X_0$ over $Y_0$ and let $f^s:X^s_0\to Y_0$ be the induced morphism. We take a resolution of singularities $\rho:X^{(s)}_0\to X^s_0$ which is an isomorphism over a nonempty Zariski open set of $Y_0$. Then we can write $$\mathcal O_{X^{(s)}_0}(K_{X^{(s)}_0})=\rho^*\omega_{X^s_0}\otimes 
\mathcal O_{X^{(s)}_0}(R)$$ where $R$ is an $(f^s\circ \rho)$-vertical Cartier divisor by construction. Let $p_i:X^s_0\to X_0$ be the $i$-th projection. We put $\pi_i=p_i\circ \rho:X^{(s)}_0\to X_0$. We consider $$L^{(s)}_0:=\sum _{i=1}^s \pi_i^*L_0+kR.$$ We further assume that $f_*\mathcal O_{X_0}(L_0)$ is locally free. Then there exists a generically isomorphic injection $$f^{(s)}_*\mathcal O_{X^{(s)}_0}(L^{(s)}_0)\hookrightarrow 
\bigotimes _{i=1}^sf_*\mathcal O_{X_0}(L_0)$$ with $f^{(s)}=f^s\circ \rho$. By construction, we have $$L^{(s)}_0
\sim _{\mathbb R}k\left(K_{X^{(s)}_0/Y_0}+\sum _{i=1}^s \pi_i^*\Delta_0\right).$$ Note that $$\left(X^{(s)}_0, \sum _{i=1}^s \pi_i^*\Delta_0\right)$$ is log canonical over a nonempty Zariski open set of $Y_0$. We also note that $X^{(s)}_0$ may be reducible, that is, $X^{(s)}_0$ may be a disjoint union of smooth varieties.

By the flat base change theorem, we have $$\omega_{X^s_0/Y_0}=\bigotimes _{i=1}^s p^*_i\omega_{X_0/Y_0}.$$ In particular, $X^s_0$ is Gorenstein. We note that $$\label{x-3.7}
\begin{split}
L^{(s)}_0&=\sum _{i=1}^s \rho^*p^*_i (kK_{X_0/Y_0}+(L_0-kK_{X_0/Y_0}))+kR
\\& \sim kK_{X^{(s)}_0/Y_0}+\sum _{i=1}^s \pi^*_i (L_0-kK_{X_0/Y_0}). 
\end{split}$$

We have the following isomorphism of locally free sheaves: $$f^s_*\mathcal O_{X^s_0}\left(\sum _{i=1}^s p^*_iL_0\right)\simeq 
\bigotimes ^s f_*\mathcal O_{X_0}(L_0).$$

We use induction on $s$. If $s=1$, then the statement 