 a fundamental theorem of the theory of mixed-$\omega$-sheaves.  \[x-thm0.0\] Let $f:X\to Y$ be a surjective morphism from a normal projective variety $X$ onto a smooth projective variety $Y$. 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  -   $L+f^*D-k(K_{X/Y}+\Delta)-f^*A$ is semi-ample for some big $\mathbb R$-divisor $A$ on $Y$.  If $f_*\mathcal O_Y(L)\ne 0$, then $$\mathcal O_Y(K_Y+\lceil D\rceil)\otimes f_*\mathcal O_X(L)$$ is a mixed-$\widehat \omega$-big-sheaf on $Y$.  We divide the proof into several small steps.  \[x-step0.0