  \[x-thm0.0\] Let $\mathcal F$ be a mixed-$\omega$-sheaf on a smooth quasi-projective variety $W$. Then $\mathcal F\otimes \omega^{\otimes -0}_W$ is weakly positive.  We may assume that $\mathcal F$ is a direct summand of $f_*\mathcal O_V(K_V+D)$ as in Definition \[x-def0.0\]. 