0.0\] Let $f:X\to Y$ be a surjective morphism from a normal projective variety $X$ onto a smooth projective variety $Y$ with $\dim Y=n$. Let $L$ be a Cartier divisor on $X$ and let $\Delta$ be an effective $\mathbb R$-divisor on $X$ such that $K_X+\Delta$ is $\mathbb R$-Cartier. Let $D$ be an $\mathbb R$-divisor on $Y$. Let $k$ be a positive integer with $k\geq 0$. Assume the following conditions:  -   $(X, \Delta)$ is log canonical over a nonempty Zariski open set of $Y$, and  -   $L+f^*D-k(K_{X/Y}+\Delta)$ is nef and $f$-semi-ample.  Let $H$ be a big Cartier divisor on $Y$ such that $|H|$ is free. Then $$