that $x^{n-s'}f(x)\in x^{-2}U[[x^{-1}]]$, so that $$\begin{aligned}
\Res_{x}x^{n-k-i}(x-z)^{k}f(x)=0\end{aligned}$$ for $i\ge s'$. Then use $(x-z)^{-k}$ instead of $(-z+x)^{-k}$ in (\[e3.30f\]) to complete the proof. $\;\;\;\;\;\Box$

In view of Lemmas \[ltruncationyr\] and \[lbasic5\] we immediately have:

Let $v\in V,\; \alpha\in {\cal{D}}_{P(z)}(W)$ and let $k\in {\N}$ be such that (\[e3.11\]) holds. Then for $n\in {\Z}$, $$\begin{aligned}
\label{relate}
v^{R}_{n}\alpha
\in \mbox{\rm linear span }\{v^{*}_{m}\alpha\;|\; m\ge n\}.\end{aligned}$$

As an immediate consequence of Corollary \[lbasic2\], Proposition \[pbasic1\] and Lemma \[ltruncationyr\] we have:

Let $v\in V,\; \alpha\in {\cal{D}}_{P(z)}(W)$. Then $$\begin{aligned}
Y^{R}(v,x)\alpha\in \left({\cal{D}}_{P(z)}(W)\right)((x)).\end{aligned}$$

$$\begin{aligned}
\label{edef2}
Y^{R}(v,x)\alpha=(z-x)^{-k}\left[(z-x)^{k}Y^{*}(v,x)\alpha\right]
\in \left({\cal{D}}_{P(z)}(W)\right)((x)).\end{aligned}$$

By Remark \[rcontain\], $D(W)$ is contained in the intersection of all ${\cal{D}}_{P(z)}(W)$ for $z\in {\C}^{\times}$. Conversely, let $$\alpha\in {\cal{D}}_{P(z_{1})}(W)\cap {\cal{D}}_{P(z_{2})}(W),$$ where $z_{1},z_{2}$ are two distinct nonzero complex numbers. Then, for $v\in V,\; w\in W$, the formal series $\<\alpha,Y^{o}(v,x)w\>$ converges in the domain $|x|>{\rm max}\{|z_{1}|,|z_{2}|\}$ to a rational function $h(x)$ such that $$h(x)\in {\C}[x,x^{-1},(x-z_{1})^{-1}]\cap {\C}[x,x^{-1},(x-z_{2})^{-1}].$$ Consequently, $h(x)\in {\C}[x,x^{-1}]$. By Remark \[rdw\], $\alpha\in D(W)$. Thus we have proved the following simple fact:

Let $W$ be a weak $V$-module. Then $$\begin{aligned}
\cap_{z\in {\C}^{\times}}{\cal{D}}_{P(z)}(W)=D(W).\;\;\;\;\Box\end{aligned}$$

As our first key result we have:

The pair $({\cal{D}}_{P(z)}(W),Y^{R})$ carries the structure of a weak $V$-module.

The truncation condition: $Y^{R}(v,x)\alpha\in {\cal{D}}_{P(z)}(W)((x))$ for $v\in V,\;\alpha\in {\cal{D}}_{P(z)}(W)$ holds (Proposition \[prd\]) and $Y^{R}({\bf 