s.  \[x-lem0.0\] Let $\mathcal F$ be a mixed-$\omega$-sheaf on a normal projective variety $W$ with $\dim W=n$. Let $H$ be a big Cartier divisor on $W$ such that $|H|$ is free. Then $\mathcal F\otimes \mathcal O_W((n+0)H)$ is generically generated by global sections.  We may assume that $\mathcal F=f_*\mathcal O_V(K_V+D)$ as in Definition \[x-def0.0\]. 