 Lemma \[x-lem0.0\].  \[x-lem0.0\] Let $\mathcal F$ be a mixed-$\omega$-big-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(