\label{ejac2}
& &{\rm Res}_{x_{0}}x_{0}^{m}
x_{0}^{-1}\delta\left(\frac{x_{1}-x_{2}}{x_{0}}\right)
\<Y^{R}(u,x_{1})Y^{R}(v,x_{2})\alpha,w\>
\nonumber\\
& &-{\rm Res}_{x_{0}}x_{0}^{m}
x_{0}^{-1}\delta\left(\frac{x_{2}-x_{1}}{-x_{0}}\right)
\<Y^{R}(v,x_{2})Y^{R}(u,x_{1})\alpha,w\>\nonumber\\
&=&{\rm Res}_{x_{0}}x_{0}^{m}
x_{2}^{-1}\delta\left(\frac{x_{1}-x_{0}}{x_{2}}\right)
\<Y^{R}(Y(u,x_{0})v,x_{2})\alpha,w\>.\end{aligned}$$ Since there is no term depending on $m$ in (\[ejac2\]) except for $x_{0}^{m}$ and $m$ is an arbitrarily fixed integer, dropping $\Res_{x_{0}}x_{0}^{m}$ from (\[ejac2\]) we obtain the desired Jacobi identity for $Y^{R}$. This concludes the proof. $\;\;\;\;\Box$

Since $Y^{R}$ depends on $z$, it is necessary, especially in certain situations, for us to use the notation $Y^{R}_{P(z)}$ for $Y^{R}$, and we shall do so from now on.

Recall that given a left weak $V$-module $(W,Y_{W})$ and a nonzero complex number $z$, we have a right weak $V$-module $(W,(Y^{o}_{W})^{(z)})$ (Propositions \[pcontragredient\] and \[ptranslationz\]). Using this and the linear maps $\iota_{x;\infty}$ and $\iota_{x;0}$ defined earlier we now define another vertex operator map $Y_{P(z)}^{L}$; a linear map from $V$ to $(\End\; {\cal{D}}_{P(z)}(W))[[x,x^{-1}]]$.

Since $\<\alpha,Y^{o}(v,y)w\>$ absolutely converges in the domain $|y|>|z|$ to a rational function in ${\C}[y,y^{-1},(y-z)^{-1}]$, the formal series $\<\alpha,Y^{o}(v,x+z)w\>$ absolutely converges in the domain $|x|>|z|$, $|x+z|>|z|$ to a rational function in ${\C}[x,x^{-1},(x+z)^{-1}]$, where the possible poles at $y=0, z$ are transformed to $x=-z,0$. In particular, $\<\alpha,Y^{o}(v,x+z)w\>$ absolutely converges in the domain $|x|>2|z|$ (to the same rational function). Thus, in view of Remark \[rcondomain\] and Definitions \[dleftaction\] and 