$f=\sum_{i=1}^n a_ix_i$.

By lemma \[lem:BernsteinVFilt\], $b_h(s)$ is the minimal polynomial of $-\partial_t t$ on $
\frac{{\mathcal{D}}_V[t\partial_t]\delta(t-h)}{t{\mathcal{D}}_V[t\partial_t]\delta(t-h)}
$. In particular, by theorem \[theo:BernsteinPolLinFree\], the functional equation $$\left(t\cdot h^*(\partial_{x_i})-b_h(-\partial_t t)\right)\delta(t-h)=0$$ holds in $(i_h)_+{\mathcal{O}}_V(*D)$. Then it follows directly from the definition of the twisted module ${\mathcal{O}}_V(*D)e^{-f}$ that the functional equation from above holds in $(i_h)_+{\mathcal{O}}_V(*D)e^{-f}$. Now suppose that there is another equation $$\left(t\cdot \widetilde{P}(x_i,\partial_{x_i},-\partial_t t)-\widetilde{B}(-\partial_t t)\right)\delta(t-h)e^{-f}=0,$$ where $\widetilde{P}\in{\mathcal{D}}_V[s]$ and $\widetilde{B}(s)\in{{\mathds C}}[s]$ with $\deg(\widetilde{B})<\deg(b_h)$. Then we obtain the equation $$\left(t\cdot \widetilde{P}(x_i,\partial_{x_i}-a_i,-\partial_t t)-\widetilde{B}(-\partial_t t)\right)\delta(t-h)=0$$ in $(i_h)_+{\mathcal{O}}_V(*D)$, which contradicts the minimality of $b_h(s)$.

We consider, as in the last lemma, the ${\mathcal{D}}_{T\times V}$-module $$(i_h)_+{\mathcal{O}}_V(*D)e^{-f}\cong (i_h)_*{\mathcal{O}}_V(*D)e^{-f}[\partial_t]$$ and the ${\mathcal{D}}_V[t\partial_t]$-submodule generated (over ${\mathcal{D}}_V[t\partial_t]$) by $\delta(t-h)e^{-f}$. The direct image $h_+({\mathcal{O}}_V(*D)e^{-f})$ is obtained in the standard way from $(i_h)_+{\mathcal{O}}_V(*D)e^{-f}$ as the relative de Rham complex of the projection $p_1:T\times V\rightarrow T$. In other words, we have $${\mathcal{H}}^i(h_+({\mathcal{O}}_V(*D)e^{-f}))  =  {\mathcal{H}}^i((p_1)_*\operatorname{\mathit{DR}}^{n+\bullet}_{T\times V/T}((i_h)_*{\mathcal{O}}_V(*D)e^{-f}[\partial_t]))$$ Considering $(i_h)_+{\mathcal{O}}_V(*D)e^{-f}$ as a ${\mathcal{D}}_V$-module only, we thus have $${\mathcal{H}}^i(h_+({\mathcal{O}}_V(*D)e^{-f}))=h_*{\mathcal{H}}^i(\operatorname{\mathit{DR}}^{n+\bullet}_V{\mathcal{O}}_V(*D)e^{-f}[\partial_t]).$$ Now it is well known (see, e.g., [@Mal6 proposition 2.1] or [@Bj2 proposition 2.2.10]), that for any left ${\mathcal{D}}_V$-module ${\mathcal{L}}$, the de Rham complex $\operatorname{\mathit{DR}}_V^\bullet({\mathcal{L}})$ represents the (shifted) derived tensor product $\Omega^n_V\stackrel{{{\mathbb L}}}{\otimes}_{{\mathcal{D}}_V}{\mathcal{L}}[-n]$, in particular, we have $${\mathcal{H}}^n(DR^\bullet_V({\mathcal{L}})) \cong \Omega^n_V\otimes_{{\mathcal{D}}_V}{\mathcal{L}}.$$ It follows that $$\label{eq:TopDeRham}
{\mathcal{H}}^0(h_+({\mathcal{O}}_V(*D)e^{-f})) \cong h_*\left(\Omega^n_V\otimes_{{\mathcal{D}}_V}({\mathcal{O}}_V(*D)e^{-f}[\partial_t])\right).$$ so that, taking 