\Phi (v_{1}, x_{1};  \ldots; v_{k}, x_{k}), \zeta_1\right)\; u   \rangle
\nn
& &
\qquad   \qquad %\cdot
\langle w', Y^{W}_{WV}\left(
\Psi
(v'_{1}, y_{1}; \ldots; v'_{n}, y_{n}) , \zeta_2\right) \; \overline{u} \rangle
\end{eqnarray*}

\begin{eqnarray*}
=  \sum_{l\in \mathbb{Z}
} \epsilon^{l} \sum_{u\in V_l}
\langle w', e^{\zeta_1 \; L_W{(-1)}} \; Y_{W} \left( u, -\zeta_1\right) \;
\Phi (v_{1}, x_{1};  \ldots; v_{k}, x_{k})  \rangle
\nn
& &
\qquad   \qquad
\langle w', e^{\zeta_2 \; L_W{(-1)}} \; Y_{W} \left( \overline{u},  -\zeta_2 \right) \;
\Psi
(v'_{1}, y_{1}; \ldots; v'_{n}, y_{n}) \rangle.
\end{eqnarray*}

Recall from (<ref>) (see Appendix <ref>) that in two sphere $\epsilon$-sewing formulation,

the complex parameters $\zeta_a$, $a=1$, $2$ are coordinates inside identified annuluses ${\mathcal A}_a$, and

$|\zeta_a|\le r_a$.

Therefore, due to Proposition <ref>

the matrix elements

\begin{eqnarray}
\label{format1}
\widetilde{\mathcal R}(x_{1},  \ldots, x_{k}; \zeta_1 ) &=&
\langle w',
e^{\zeta_1 \; L_W{(-1)}} \; Y_{W} \left( u, -\zeta_1\right) \;
\Phi (v_{1}, x_{1};  \ldots; v_{k}, x_{k})   \rangle,
\end{eqnarray}

\begin{eqnarray}
\label{format2}
\widetilde{\mathcal R}(y_{1}, \ldots, y_{n}; \zeta_2)&=& \langle w',
e^{\zeta_2 \; L_W{(-1)}} \; Y_{W} \left( \overline{u},  -\zeta_2 \right) \;
\Psi
(v'_{1}, y_{1}; \ldots; v'_{n}, y_{n})   \rangle,
\end{eqnarray}

are absolutely convergent in powers of $\epsilon$ with some radia of convergence $R_a\le r_a$, with
$|\zeta_a|\le R_a$.

The dependence of (<ref>) and (<ref>) on $\epsilon$ is expressed via $\zeta_a$, $a=1$, $2$.

Let us rewrite the product (<ref>) as

\begin{eqnarray}
\label{pihva}
&& \langle w',
\Theta(v_1, x_1; \ldots; v_{k}, x_{k}; v'_1, y_1; \ldots; v'_{n}, y_{n};
\epsilon
\nn
\qquad =
\sum\limits_{l\in Z} \epsilon^l \left(\langle w',
\Theta (v_1, x_1; \ldots; v_{k}, x_{k}; v'_1, y_1; \ldots; v'_{n}, y_{n}
\right)\rangle)_l
\nn
=  \sum_{l\in \mathbb{Z}}
\sum_{u\in V_l} \sum\limits_{m \in \C}
\epsilon^{l-m-1} \;  \widetilde{\mathcal R}_m(x_1, \ldots, x_{k}; \zeta_1) \;
\widetilde{\mathcal R}_m(y_1, \ldots, y_{n}; \zeta_2),
\end{eqnarray}

as a formal series in $\epsilon$

for $|\zeta_{a}|\leq R_{a}$, where

$|\epsilon |\leq r$ for $r<r_{1}r_{2}$.

Then we

Cauchy's inequality to

coefficient forms (<ref>) and (<ref>)

to find

\begin{equation}
\left| \widetilde{\mathcal R}_{m}
(x_{1}, \ldots, x_{k}; \zeta_1) \right| \leq {M_1} {R_1^{-m}},  \label{Cauchy1}
\end{equation}

\[
M_1=\sup_{ \left| \zeta_{1} \right| \leq R_{1}, \left|\epsilon \right| \leq r}
\left| \widetilde{\mathcal R}(x_{1}, \ldots, x_{k}; \zeta_1 ) \right|.
\]

\begin{equation}
\left|  \widetilde{\mathcal R}_{m} (y_{1}, \ldots, y_{n}; \zeta_2) \right
| \leq {M_2} {R_2^{-m}},  \label{Cauchy2}
\end{equation}

\[
M_2=\sup_{|\zeta_{2}|\leq R_{2},|\epsilon |\leq r} \left|\widetilde{\mathcal R}(y_{1}, \ldots, y_{n}; \zeta_2)\right|.
\]

Using (<ref>) and (<ref>) we obtain for (<ref>)

\begin{eqnarray}
\label{borona}
\left|
\left(\langle w',
\F (v_1, x_1; \ldots; v_{k}, x_{k}; v'_1, y_1; \ldots; v'_{n}, y_{n}
\rangle \right)_l\right|
\nn
\qquad \le
\left|\widetilde{\mathcal R}_{m} (x_1, \ldots, x_{k}; \zeta_1)\right|
\; \left|\widetilde{\mathcal R}_{m}(y_{1}, \ldots, y_{n}; \zeta_2)\right|
\nn
\qquad  \le
M_1 \; M_2 \; \left(R_1
\end{eqnarray}

Thus, for

$M=\min\left\{M_1, M_2\right\}$ and $R=\max\left\{R_1, R_2\right\}$,

such that

\begin{equation}
\left|
\mathcal R_{l}(
x_{1};  \ldots, x_{k};
y_{1},  \ldots, y'_{n}
; \zeta_1, \zeta_2)
\right| \leq
{M} {R^{-l+m+1}}.  \label{Cauchy}
\end{equation}

Thus, we see that (<ref>) is absolute convergent

as a formal series in $\epsilon$

is defined

for $|\zeta_{a}|\leq r_{a}$, and

$|\epsilon |\leq r$ for $r<r_{1}r_{2}$,

with extra poles

only at $x_i=y_j$, $1\le i \le k$, $1\le j \le n$.

§.§ Proof of Proposition <ref>

By using (<ref>) for $\Phi(v_{1},  x_{1};  \ldots; v_{k}, x_{k})$

and $\Psi(v'_{1},  y_{1}; \ldots;  v'_{n},  y_{n})$,

we consider

\begin{eqnarray}
\label{Z2n_pt_eps1q00000}
& &
\langle w',\partial_{
l} %
\Theta( v_{1},  x_{1};  \ldots;  v_k, x_{k};   v'_1, y_1; \ldots;  v'_{n},  y_{n}; \epsilon) \rangle
\nn
\nn
& &  = \sum_{l\in \mathbb{Z}
} \epsilon^{l} \sum_{u\in V_l}
\langle w', \partial^{\delta_{l,i}}_{x_i}
\Phi ( v_{1},  x_{1};  \ldots; v_{k}, x_{k}), \zeta_1 \right) \; u \rangle
\nn
& &
\qquad   \qquad
\langle w', \partial^{\delta_{l,j}}_{y_j}
\Psi
(v'_{1},  y_{1}; \ldots;  v'_{n},  y_{n}),  \zeta_2 \right) \overline{u} \rangle
\nn
& &  = \sum_{l\in \mathbb{Z}
} \epsilon^{l} \sum_{u\in V_l}
\langle w', \partial^{\delta_{l,i}}_{x_i}
Y_{W}\left( u, - \zeta_1 \right)
\Phi ( v_{1},  x_{1};  \ldots; v_{k}, x_{k})) \; u \rangle
\nn
& &
\qquad   \qquad %\cdot
\langle w', \partial^{\delta_{l,j}}_{y_j}
Y_{W}\left(\overline{u}, - \zeta_2  \right)
\Psi
(v'_{1},  y_{1}; \ldots;  v'_{n},  y_{n})  \rangle
\nn
& &  = \sum_{l\in \mathbb{Z}}
\epsilon^{l} \sum_{u\in V_l}
\langle w',
\partial^{\delta_{l,i}}_{x_i}  \Phi ( v_{1},  x_{1};  \ldots; v_{k}, x_{k}), \zeta_1 \right) \; u \rangle
\nn
& &
\qquad   \qquad %\cdot
\langle w',
Y^{W}_{WV}\left( \partial^{\delta_{l,j}}_{y_j}
\Psi
(v'_{1},  y_{1}; \ldots;  v_{n},  y_{n}),  \zeta_2 \right) \overline{u} \rangle
\nn
& &
=  \sum_{l\in \mathbb{Z}
} \epsilon^{l} \sum_{u\in V_l}
\langle w',
\Phi (v_{1}, x_{1}; \ldots; \left(L_V{(-1)}\right)^{\delta_{l,i}} v_i, x_i; \ldots; v_{k}, x_{k}), \zeta_1 \right) \; u \rangle
\nn
& &
\qquad   \qquad %\cdot
\langle w', Y^{W}_{WV}\left(
\Psi
(v'_{1}, y_{1}; \ldots; \left(L_V{(-1)}\right)^{\delta_{l,j}} v'_j, y_j;   \ldots; v'_{n}, y_{n}),  \zeta_2 \right) \overline{u} \rangle
\nn
& &
\langle w',
\Theta (
v_{1},  x_{1};  \ldots;  \left(L_V{(-1)}\right)_l;  \ldots;  v'_{n},  y_{n};
\epsilon) \rangle,
\end{eqnarray}

where $\left(L_V{(-1)}\right)_l$ acts on the $l$-th entry of

$(v_{1},  \ldots;  v_k;   v'_1,  \ldots,   v'_{n})$.

Summing over $l$ we obtain

\begin{eqnarray}
\label{lder2f0}
\sum\limits_{l=1}^{k+n} \partial_{
\Theta (v_{1}, x_{1}; \ldots;
v_{k}, x_{k};
v'_{1}, y_{1}; \ldots;
v'_{n}, y_{n}; \epsilon) \rangle
\nn
\sum\limits_{l=1}^{k+n} \langle w',
\Theta (v'_{1}, x_{1}; \ldots; \left(L_V{(-1)}\right);    \ldots;  v'_{n}, y_{n}; \epsilon) \rangle
\nn
=\langle w',
L_{W}{(-1)}.\Theta(v_{1}, x_{1};  \ldots ; v_{k}, x_{k};   v'_{1},  y_{1}; \ldots; v'_{n},  y_{n}; \epsilon) \rangle.
\end{eqnarray}

Due to (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>),

we have

\begin{eqnarray*}
&& \langle w',
\Theta ( z^{L_V{(0)} } v_{1}, z \; x_{1};  \ldots;  z^{L_V{(0)}} v_{k},  z\; x_{k};
z^{L_V{(0)}} v'_{1}, z \; y_{1};  \ldots;  z^{L_V{(0)}} v'_{n},  z\; y_{n};
\epsilon) \rangle
\end{eqnarray*}

\begin{eqnarray*}
& &  = \sum_{l\in \mathbb{Z}
} \epsilon^{l} \sum_{u\in V_l}
\langle w', Y^{W}_{WV}\left(
\Phi (z^{L_V{(0)}} v_{1}, z\; x_{1};  \ldots; z^{L_V{(0)}} v_{k}, z\; x_{k}), \zeta_1 \right) \; u \rangle
\nn
& &
\qquad   \qquad %\cdot
\langle w', Y^{W}_{WV}\left(
\Psi
(z^{L_V{(0)}} v'_{1}, z \; y_{1}; \ldots; z^{L_V{(0)}} v'_{n}, z\; y_{n}),  \zeta_2 \right) \overline{u} \rangle
\end{eqnarray*}

\begin{eqnarray*}
& &
=  \sum_{l\in \mathbb{Z}
} \epsilon^{l} \sum_{u\in V_l}
\langle w', Y^{W}_{WV}\left( z^{L_V{(0)}}
\Phi (v_{1}, x_{1};  \ldots; v_{k}, x_{k}), \zeta_1 \right) \; u \rangle
\nn
& &
\qquad   \qquad %\cdot
\langle w', Y^{W}_{WV}\left( z^{L_V{(0)}}
\Psi
(v'_{1}, y_{1}; \ldots; v'_{n}, y_{n}),  \zeta_2 \right) \overline{u} \rangle
\end{eqnarray*}

\begin{eqnarray*}
& &
=  \sum_{l\in \mathbb{Z}
} \epsilon^{l} \sum_{u\in V_l}
\langle w', e^{\zeta_1 L_W{(-1)}}  Y_{W} \left(  u, -\zeta_1 \right)
\Phi (v_{1}, x_{1};  \ldots; v_{k}, x_{k})  \rangle
\nn
& &
\qquad   \qquad %\cdot
\langle w', e^{\zeta_2 L_W{(-1)}} \; Y_{W}\left( \overline{u}, -\zeta_2  \right) z^{L_V{(0)}}
\Psi
(v'_{1}, y_{1}; \ldots; v'_{n}, y_{n})   \rangle
\end{eqnarray*}

\begin{eqnarray*}
& &
=  \sum_{l\in \mathbb{Z}
} \epsilon^{l} \sum_{u\in V_l}
\langle w', e^{\zeta_1 L_W{(-1)}}  z^{L_V{(0)}} Y_{W} \left(  z^{-L_V{(0)}} u, -z \; \zeta_1 \right)
\Phi (v_{1}, x_{1};  \ldots; v_{k}, x_{k})  \rangle
\nn
& &
\qquad   \qquad %\cdot
\langle w', e^{\zeta_2 L_W{(-1)}}\; z^{L_W{(0)}}\;   Y_{W}\left( z^{-L_V{(0)}} \overline{u}, -z \; \zeta_2  \right)
\Psi
(v'_{1}, y_{1}; \ldots; v'_{n}, y_{n})   \rangle
\nn
& &
=  \sum_{l\in \mathbb{Z}
} \epsilon^{l} \sum_{u\in V_l}
\langle w', e^{\zeta_1 L_W{(-1)}}  z^{L_W{(0)}} z^{-{\rm wt} u} \; Y_{W} \left(   u, -z \; \zeta_1 \right)
\Phi (v_{1}, x_{1};  \ldots; v_{k}, x_{k})  \rangle
\nn
& &
\qquad   \qquad %\cdot
\langle w', e^{\zeta_2 L_W{(-1)}}\; z^{L_W(0)}\;  z^{-{\rm wt}\overline{u} } \; Y_{W}\left(  \overline{u}, -z \; \zeta_2  \right)
\Psi
(v'_{1}, y_{1}; \ldots; v'_{n}, y_{n})   \rangle
\end{eqnarray*}

\begin{eqnarray*}
& &
=  \sum_{l\in \mathbb{Z}
} \epsilon^{l} \sum_{u\in V_l}
\langle w', z^{L_W(0)} e^{\zeta_1 L_W{(-1)}} Y_{W} \left(  u, -z \zeta_1 \right)
\Phi (v_{1}, x_{1};  \ldots; v_{k}, x_{k})  \rangle
\nn
& &
\qquad   \qquad %\cdot
\langle w', z^{L_W(0)} e^{\zeta_2 L_W{(-1)}}  Y_{W}\left( \overline{u}, -z \zeta_2  \right)
\Psi
(v'_{1}, y_{1}; \ldots; v'_{n},y_{n}),   \rangle
\end{eqnarray*}

\begin{eqnarray*}
& &
=  \sum_{l\in \mathbb{Z}
} \epsilon^{l} \sum_{u\in V_l}
\langle w', z^{L_W(0)} \; Y^{W}_{WV}\left(
\Phi (v_{1}, x_{1};  \ldots; v_{k}, x_{k}), z \zeta_1 \right) \; u \rangle
\notag
\nn
& &
\qquad   \qquad %\cdot
\langle w', z^{L_W(0)} \;  Y^{W}_{WV}\left(
\Psi
(v'_{1}, y_{1}; \ldots; v'_{n},y_{n}),  z \zeta_2 \right) \overline{u} \rangle
\end{eqnarray*}

\begin{eqnarray*}
& &
=  \sum_{l\in \mathbb{Z}
} \epsilon^{l} \sum_{u\in V_l}
\langle w', z^{L_W(0)} \; Y^{W}_{WV}\left(
\Phi (v_{1}, x_{1};  \ldots; v_{k}, x_{k}),  \zeta'_1 \right) \; u \rangle
\nn
& &
\qquad   \qquad %\cdot
\langle w', z^{L_W(0)} \;  Y^{W}_{WV}\left(
\Psi
(v'_{1}, y_{1}; \ldots; v'_{n},y_{n}),  \zeta'_2 \right) \overline{u} \rangle
\end{eqnarray*}

\begin{eqnarray*}
=\langle w', \left( z^{L_W(0)} \right).
\Theta (v_{1}, x_{1};  \ldots;  v_{k}, x_{k}; v'_{1}, y_{1}; \ldots; v'_{n}, y_{n}; \epsilon) \rangle.
\end{eqnarray*}

With (<ref>),

we obtain (<ref>) for (<ref>).

§.§ Proof of Proposition <ref>

Note that due to Proposition <ref>

\begin{eqnarray*}
\Phi (v_{1}, x'_{1};  \ldots; v_{k}, x'_{k}) &=&
\Phi (v_{1}, x_{1};  \ldots; v_{k}, x_{k}),
\nn
\Psi (v_{1}, y'_{1};  \ldots; v_{n}, y'_{n}) &=&
\Psi (v_{1}, y_{1};  \ldots; v_{n}, y_{n}).
\end{eqnarray*}

\begin{eqnarray*}
&& \langle w', \Theta(v_1, x'_1; \ldots,;  v_k, x'_k; v'_1, y'_1; \ldots; v'_n, y'_{n}; \epsilon) \rangle
\nn
& &  =  \sum_{l\in \mathbb{Z}
} \epsilon^{l} \sum_{u\in V_l}
\langle w', Y^{W}_{WV}\left(
\Phi (v_{1}, x'_{1};  \ldots; v_{k}, x'_{k}), \zeta_1 \right) \; u \rangle
\nn
& &
\qquad   \qquad %\cdot
\langle w', Y^{W}_{WV}\left(
\Psi
(v'_{1}, y'_{1}; \ldots; v'_{n}, y'_{n}),  \zeta_2 \right) \overline{u} \rangle
\nn
& &
=  \sum_{l\in \mathbb{Z}
} \epsilon^{l} \sum_{u\in V_l}
\langle w', Y^{W}_{WV}\left(
\Phi (v_{1}, x_{1};  \ldots; v_{k}, x_{k}), \zeta_1 \right) \; u \rangle
\nn
& &
\qquad   \qquad %\cdot
\langle w', Y^{W}_{WV}\left(
\Psi
(v'_{1}, y_{1}; \ldots; v'_{n}, y_{n}),  \zeta_2 \right) \overline{u} \rangle
\nn
\langle w', \Theta(v_1, x_1; \ldots,;  v_k, x_k; v'_1, y_1; \ldots; v'_n, y_{n}; \epsilon) \rangle.
\end{eqnarray*}

Thus, the product (<ref>) is invariant under (<ref>).

§.§ Proof of Lemma <ref>

Let $\widetilde{v}_{i} \in V$, $1 \le i \le k$,

$\widetilde{v}_{j} \in V$, $1 \le j \le k$,

$z_j$ are corresponding formal parameters.

We show that

the $\epsilon$-product of

$\Phi(\widetilde{v}_{1}, z_1;  \ldots;  \widetilde{v}_{k}, z_k)$ and

$\Psi(\widetilde{v}_{k+1}, z_{k+1}$;   $\ldots$; $\widetilde{v}_{n}, z_n)$, i.e.,

the $\W_{z_1, \ldots, z_{k+n-r}}$-valued

differential form

\begin{eqnarray}
& & {\Theta}
((\widetilde{v}_{1}, z_1;  \ldots;  \widetilde{v}_{k}, z_k);
(\widetilde{v}_{k+1}, z_{k+1};  \ldots; \widetilde{v}_{n}, z_n);
\zeta_{1}, \zeta_{2}; \epsilon )
\end{eqnarray}

is independent of the choice of $0 \le k \le n$.

\begin{eqnarray}
\label{ishodnoe}
\langle w',
\Theta (\widetilde{v}_{1}, z_{1};\ldots ;  \widetilde{v}_{k}, z_{k}; \widetilde{v}_{k+1}, z_{k+1};
\ldots ; \widetilde{v}_{n}, z_{n}; \zeta_1, \zeta_2; \epsilon)  \rangle
\nn
&&\qquad  =
\sum_{l\in \mathbb{Z}
} \epsilon^{l} \sum_{u \in V_l}
\langle w', Y^{W}_{WV}\left(
\Phi (\widetilde{v}_{1}, z_{1};  \ldots; \widetilde{v}_{k}, z_{k}), \zeta_1 \right) \; u \rangle
\nn
& &
\qquad   \qquad  \qquad %\cdot
\langle w', Y^{W}_{WV}\left(
\Psi
(\widetilde{v}_{k+1}, z_{k+1}; \ldots; \widetilde{v}_{n}, z_{n}),  \zeta_2 \right) \overline{u} \rangle.
\end{eqnarray}

On the other hand, for $0 \le m \le k$,  consider

\begin{eqnarray*}
\sum_{l\in \mathbb{Z}
} \epsilon^{l} \sum_{u \in V_l}
\langle w', Y^{W}_{WV}\left(
\Phi (\widetilde{v}_{1}, z_{1};  \ldots; \widetilde{v}_{m}, z_{m}), \zeta_1 \right) \; u \rangle
\nn
& &
\qquad   \qquad  \qquad %\cdot
\langle w', Y^{W}_{WV}\left(
\Psi
(   \widetilde{v}_{m+1}, z'_{m+1}; \ldots; \widetilde{v}_{k}, z'_{k};
\widetilde{v}_{k+1}, z_{1}; \ldots; \widetilde{v}_{n}, z_{n}),  \zeta_2 \right) \overline{u} \rangle
\nn
\qquad =\langle w',
\Theta (\widetilde{v}_{1}, z_{1};\ldots ;  \widetilde{v}_{m}, z_{m};
\widetilde{v}_{m+1}, z'_{m+1}; \ldots; \widetilde{v}_{k}, z'_{k};
\widetilde{v}_{k+1}, z_{k+1};
\ldots ; \widetilde{v}_{n}, z_{n})  \rangle.
\end{eqnarray*}

The last is the $\epsilon$-product (<ref>) of

$\Phi (\widetilde{v}_{1}, z_{1};  \ldots; \widetilde{v}_{m}, z_{m})  \in
\W_{z_{1},  \ldots, z_{m}}$ and  $\Psi
(\widetilde{v}_{m+1}, z'_{m+1} $; $ \ldots $; $ \widetilde{v}_{k}, z'_{k} $; $
\widetilde{v}_{k+1}, z_{1} $; $ \ldots $;  $\widetilde{v}_{n}, z_{n}) \in \W_{z'_{m+1}, \ldots,  z'_{k};
z_{1}, \ldots, z_{n}}$.

Let us apply the invariance with respect to a subgroup of

$\left({\rm Aut}\; \Oo^{(1)}\right)^{\times (k+n)}_{z_1, \ldots, z_{k+n}}$,

with $(z_1, \ldots, z_m)$ and $(z_{k+1}, \ldots, z_n)$ remaining unchanged.

Then we obtain the same product (<ref>).

§.§ Proof of Lemma <ref>

For arbitrary $w' \in W'$, we have

\begin{eqnarray*}
\sum_{\sigma\in J_{k+n; s}^{-1}}(-1)^{|\sigma|}
\langle w',
\Theta\left(v_{\sigma(1)}, x_{\sigma(1)}; \ldots; v_{\sigma(k)}, x_{\sigma(k)};
v'_{\sigma(1)},  y_{\sigma(1)};  \ldots; v'_{\sigma(n)},  y_{\sigma(n)})\right)
\rangle
\nn
\nn
\sum_{\sigma\in J_{k+n; s}^{-1}}(-1)^{|\sigma|}  \;  %\sum\limits_{u\in V}
\sum_{l \in \Z }  \epsilon^l
\sum_{u\in V_l }
\langle w', Y^{W}_{WV}
\left(\Phi(v_{\sigma(1)}, x_{\sigma(1)}; \ldots; v_{\sigma(k)},  x_{\sigma(k)}), \zeta_1 \right) u
\rangle
\nn
\qquad \qquad \qquad  \langle w', Y^{W}_{WV}
\left(\Psi(v'_{\sigma(1)}, y_{\sigma(1)}; \ldots; v'_{\sigma(n)},  y_{\sigma(n)}), \zeta_2
\right) \overline{u} \rangle
\nn
\sum_{l \in \Z }  \epsilon^l
\sum_{u\in V_l }
\sum_{\sigma\in J_{k+n; s}^{-1}}(-1)^{|\sigma|}
\langle w', e^{\zeta_1 L_W{(-1)}} \; Y_{W}(u, -\zeta_1) \;
\Phi(v_{\sigma(1)}, x_{\sigma(1)}; \ldots; v_{\sigma(k)},  x_{\sigma(k)})
\rangle
\nn
\qquad \qquad \qquad  \langle w',  e^{\zeta_2 L_W{(-1)}} \; Y_{W}(\overline{u}, -\zeta_2) \;
\Psi(v'_{\sigma(1)}, y_{\sigma(1)}; \ldots; v'_{\sigma(n)},  y_{\sigma(n)})
\rangle
\nn
=%\sum\limits_{u\in V}
\sum_{l \in \Z }  \epsilon^l
\sum_{u\in V_l }
\langle w', e^{\zeta_1 L_W{(-1)}} \; Y_{W}(u, -\zeta_1) \;
\sum_{\sigma\in J_{k; s}^{-1}}(-1)^{|\sigma|} \Phi(v_{\sigma(1)}, x_{\sigma(1)}; \ldots; v_{\sigma(k)},  x_{\sigma(k)})
\rangle
\nn
\qquad \qquad  \langle w',  e^{\zeta_2 L_W{(-1)}} \; Y_{W}(\overline{u}, -\zeta_2) \;
\Psi(v'_{\sigma(1)}, y_{\sigma(1)}; \ldots; v'_{\sigma(n)},  y_{\sigma(n)})
\rangle
\nn
\sum_{l \in \Z }  \epsilon^l
\sum_{u\in V_l }
\langle w', e^{\zeta_1 L_W{(-1)}} \; Y_{W}(u, -\zeta_1) \;
\Phi(v_{\sigma(1)}, x_{\sigma(1)}; \ldots; v_{\sigma(k)},  x_{\sigma(k)})
\rangle
\nn
\qquad   \langle w',  e^{\zeta_2 L_W{(-1)}} \; Y_{W}(\overline{u}, -\zeta_2) \;
\sum_{\sigma\in J_{n; s}^{-1}}(-1)^{|\sigma|}  \Psi(v'_{\sigma(1)}, y_{\sigma(1)}; \ldots; v'_{\sigma(n)},  y_{\sigma(n)})
\rangle=0,
\end{eqnarray*}

_{k+n; s}= J^{-1}
_{k;s} \times J^{-1}

due to the fact that

$\F(v_{1}, x_{1}; \ldots; v_{k},  x_{k})$

$\F(v'_{1}, y_{1} $; $ \ldots $; $ v'_{n},  y_{n})$

satisfy (<ref>).

§.§ Proof of Proposition <ref>

Recall that

$\Phi(v_{1}, x_{1}; \ldots; v_{k},  x_{k})$

is composable with $m$ vertex operators, and

$\Psi(v'_{1}, y_{1}; \ldots; v'_{n},  y_{n})$

is composable with $m'$ vertex operators.

For $\Phi(v_{1}, x_{1}; \ldots; v_{k},  x_{k})$ we have:

1) Let $l_{1}, \dots, l_{k}\in \Z_+$ such that $l_{1}+\ldots +l_{k}= k+m$,  and

$v_{1}, \dots, v_{k+m} \in V$, and arbitrary $w'\in W'$. Set

\begin{eqnarray}
\label{psii}
\Xi_{i}
E^{(l_{i})}_{V}(v_{k_1}, x_{k_1}- \zeta_{i};
\ldots;
v_{k_i}, x_{k_i}- \zeta_{i}
; \one_{V}),
\end{eqnarray}

\begin{eqnarray}
\label{ki}
{k_1}={l_{1}+\ldots +l_{i-1}+1}, \quad  \ldots, \quad  {k_i}={l_{1}+\ldots +l_{i-1}+l_{i}},
\end{eqnarray}

for $i=1, \dots, k$.

Then the series

\begin{eqnarray}
\label{Inms}
\mathcal I^k_m(\Phi)=
\sum_{r_{1}, \dots, r_{k}\in \Z}\langle w',
\Phi(P_{r_{1}}\Xi_{1}; \zeta_1;
\ldots; %\otimes
P_{r_k} \Xi_{k}, \zeta_{k})
\rangle,
\end{eqnarray}

is absolutely convergent  when

\begin{eqnarray}
\label{granizy1}
|x_{l_{1}+\ldots +l_{i-1}+p}-\zeta_{i}|
+ |x_{l_{1}+\ldots +l_{j-1}+q}-\zeta_{i}|< |\zeta_{i}
\end{eqnarray}

for $i$, $j=1, \dots, k$, $i\ne j$ and for $p=1,
\dots,  l_i$ and $q=1, \dots, l_j$.

There exist positive integers $N^k_m(v_{i}, v_{j})$,

depending only on $v_{i}$ and $v_{j}$ for $i, j=1, \dots, k$, $i\ne j$, such that

the sum is analytically extended to a
rational function
in $(x_{1}, \dots, x_{k+m})$,

independent of $(\zeta_{1}, \dots, \zeta_{k})$,

with the only possible poles at

$x_{i}=x_{j}$, of order less than or equal to

$N^k_m(v_{i}, v_{j})$, for $i$, $j=1, \dots, k$,  $i\ne j$.

For $\Psi(v'_{1}, y_{1}; \ldots; v'_{n},  y_{n})$  we have:

1') Let $l'_{1}, \dots, l'_{n}\in \Z_+$ such that $l'_{1}+\ldots +l'_{n}= n+m'$,

$v'_{1}, \dots, v_{n+m'} \in V$ and arbitrary $w'\in W'$.

\begin{eqnarray}
\label{psii}
\Xi'_{i'}
E^{(l'_{i'})}_{V}(v'_{k'_1}, y_{k'_1}- \zeta'_{i'};
\ldots;
v'_{k'_{i'}}, y_{k'_{i'}}- \zeta'_{i'}
; \one_{V}),
\end{eqnarray}

\begin{eqnarray}
\label{ki}
{k'_{1}}={l'_{1}+\ldots +l'_{i'-1}+1}, \quad  \ldots, \quad  {k'_{i'}}={l'_{1}+\ldots +l'_{i'-1}+l'_{i'}},
\end{eqnarray}

for $i'=1, \dots, n$.

Then the series

\begin{eqnarray}
\label{Jnms}
\mathcal I^{n}_{m'}(\Psi)=
\sum_{r'_{1}, \dots, r'_{n}\in \Z}\langle w',
\Psi(P_{r'_{1}}\Psi'_{1}; \zeta'_1;
\ldots; %\otimes
P_{r'_{n}} \Psi'_{n}, \zeta'_{n})
\rangle,
\end{eqnarray}

is absolutely convergent  when

\begin{eqnarray}
\label{granizy2}
|y_{l'_{1}+\ldots +l'_{i'-1}+p'}-\zeta'_{i'}|
+ |y_{l'_{1}+\ldots +l'_{j'-1}+q'}-\zeta'_{i'}|< |\zeta'_{i'}
\end{eqnarray}

for $i'$, $j'=1, \dots, n$, $i'\ne j'$ and for $p'=1,
\dots,  l'_i$ and $q'=1, \dots, l'_j$.

There exist positive integers $N^{n}_{m'}(v'_{i'}, v'_{j'})$,

depending only on $v'_{i'}$ and $v'_{j'}$ for $i$, $j=1, \dots, n$, $i'\ne j'$, such that

the sum is analytically extended to a
rational function

in $(y_{1}, \dots, y_{n+m'})$,

independent of $(\zeta'_{1}, \dots, \zeta'_{n})$,

with the only possible poles at

$y_{i'}=y_{j'}$, of order less than or equal to

$N^{n}_{m'}(v'_{i'}, v'_{j'})$, for $i'$, $j'=1, \dots, n$,  $i'\ne j'$.

Now let us consider the first condition of Definition <ref> of composability for the product

(<ref>) of

$\Phi(v_{1}, x_{1}; \ldots; v_{k},  x_{k})$

$\Psi(v'_{1}, y_{1}; \ldots; v'_{n},  y_{n})$ with a number of vertex operators.

Then we obtain for $\Theta\left(v_{1}, x_{1}; \ldots; v_k, x_k; v'_1, y_1;
\ldots; v'_{n}, y_{n}; \epsilon \right)$ the following.

We redefine the notations for the

\begin{eqnarray*}
&& (v''_{1}, \ldots, v''_k; v''_{k+1}, \ldots, v''_{k+m}; v''_{k+m+1}, \dots, v''_{k+n+m+m'};
v_{n+1}, \ldots, v'_{n+m'})
\nn
\qquad \qquad
=(v_{1}, \ldots, v_k; v_{k+1}, \ldots, v_{k+m}; v'_1, \dots, v'_{n};
v'_{n+1}, \ldots, v'_{n+m'}), %\in V,
\nn
(z_{1}, \ldots, z_k; z_{k+1},  \dots, z_{k+n-r}) =
(x_1, \ldots, x_k; y_1, \ldots, y_{n}),
\end{eqnarray*}

of vertex algebra $V$ elements.

Introduce $l''_{1}, \dots, l''_{k+n} \in \Z_+$,

such that $l''_{1}+\ldots +l''_{k+n}= k+n+m+m'$.

\begin{eqnarray}
\label{psiinew}
\Xi''_{i}
E^{(l''_{i''})}_{V}(v''_{k''_1}, z_{k''_1}- \zeta''_{i''};
\ldots;
v''_{k''_{i''}}, z_{k''_{i''}}- \zeta''_{i''}
; \one_{V}),
\end{eqnarray}

\begin{eqnarray}
\label{ki}
{k''_1}={l''_{1}+\ldots +l''_{i''-1}+1}, \quad  \ldots, \quad  {k''_{i''}}={l''_{1}+\ldots +l''_{i''-1}+l''_{i''}},
\end{eqnarray}

for $i''=1, \dots, k+n$,

and we take

\[
(\zeta''_1, \ldots, \zeta''_{k+n})= (\zeta_1, \ldots, \zeta_{k}; \zeta'_1, \ldots, \zeta'_{n}).
\]

Then we consider

\begin{eqnarray}
\label{Inmdvadva}
&& \mathcal I^{k+n}_{m+m'}(\Theta)=
\sum_{r''_{1}, \dots, r''_{k+n}\in \Z}
\langle w',
\Theta(P_{r''_{1}}\Psi''_{1}; \zeta''_1;
\ldots;
P_{r''_{k+n}} \Psi''_{k+n}, \zeta''_{k+n})
\rangle,
\end{eqnarray}

and prove it is absolutely convergent with some conditions.

\begin{eqnarray}
\label{granizy1000000}
|z_{l''_{1}+\ldots +l''_{i-1}+p''}-\zeta''_{i}|
+ |z_{l''_{1}+\ldots +l''_{j-1}+q''}-\zeta''_{i}|< |\zeta''_{i} -\zeta''_{j}|,
\end{eqnarray}

of absolute convergence for (<ref>) for $i''$, $j''=1, \dots, k+n$, $i\ne j$ and for $p''=1,
\dots,  l''_i$ and $q''=1, \dots, l''_j$, follows from the conditions (<ref>) and (<ref>).

The action of $e^{\zeta L_W{(-1)} } \;  Y_W(.,.)$, $a=1$, $2$, in

\[
\langle w',  e^{\zeta_1 L_W{(-1)} } \; Y_W({u}, -\zeta)\sum_{r_{1}, \dots, r_{k}\in \Z}
\Phi(P_{r_{1}}\Xi_{1}; \zeta_1;
\ldots;
P_{r_{k}} \Xi_{k}, \zeta_{k})   \rangle,
\]

\[
\langle w',  e^{\zeta_2 L_W{(-1)} } \; Y_W(\overline{u}, -\widetilde{\zeta}) \sum_{r'_{1}, \dots, r'_{n}\in \Z}
\Psi(P_{r'_{1}}\Xi'_{1}; \zeta_1;
\ldots;
P_{r'_{k}} \Xi'_{n}, \zeta'_{n})  \rangle,
\]

does not affect the absolute convergency of (<ref>) and (<ref>).

We obtain

\begin{eqnarray*}
&& \left|\mathcal I^{k+n}_{m+m'}(\Theta)\right|=
\nn
=\left|\sum_{r''_{1}, \dots, r''_{k+n}\in \Z}
\langle w',
\Theta(P_{r''_{1}}\Xi''_{1}; \zeta''_1;
\ldots;
P_{r''_{k+n}} \Xi''_{k+n}, \zeta''_{k+n})
\rangle\right|
\nn
\sum_{l \in \Z }  \epsilon^l
\sum_{u\in V_l }
\langle w',  Y^W_{VW}(\sum_{r_{1}, \dots, r_{k}\in \Z}
\Phi(P_{r_{1}}\Xi_{1}; \zeta_1;
\ldots;
P_{r_{k}} \Xi_{k}, \zeta_{k}), \zeta) u  \rangle \right.
\nn
\left.
\qquad \qquad \langle w',  Y^W_{VW}(\sum_{r'_{1}, \dots, r'_{n}\in \Z}
\Psi(P_{r'_{1}}\Xi'_{1}; \zeta'_1;
\ldots;
P_{r'_{n}} \Xi'_{n}, \zeta'_{n}), \widetilde{\zeta}) \overline{u}
\rangle \right|
\nn
\qquad \qquad \qquad \le \left|\mathcal I^{k}_{m}(\Phi)\right| \; \left|\mathcal I^{n}_{m'}(\Psi)\right|.
\end{eqnarray*}

Thus, we infer that (<ref>)

is absolutely convergent.

Recall that

the maximal orders of possible poles of (<ref>) are $N^{k}_{m}(v_{i}, v_{j})$, $N^{n}_{m'}(v'_{i'}, v'_{j'})$

at $x_{i}=x_{j}$, $y_{i'}=y_{j'}$.

From the last expression we infer that

there exist

positive integers $N^{k+n}_{m+m'}(v''_{i''}, v''_{j''})$

for $i$, $j=1, \dots, k$, $i\ne j$,   $i'$, $j'=1, \dots, n$, $i'\ne j$,

depending only on $v''_{i''}$ and $v''_{j''}$ for $i''$,  $j''=1, \dots, k+n$, $i''\ne j''$

such that

the series (<ref>)

can be analytically extended to a

rational function

in $(x_{1}, \dots, x_k; y_1, \ldots, y_n)$,

independent of $(\zeta''_{1}, \dots, \zeta''_{k+n})$,

with extra

possible poles at

and $x_{i}=y_{j}$,  of order less than or equal to

$N^{k+n}_{m+m'}(v''_{i''}, v''_{j''})$, for $i''$, $j''=1, \dots, n$,  $i''\ne j''$.

Let us proceed with the second condition of composability.

For $\Phi(v_1, x_1; \ldots; v_{k}, x_k)  \in  C^{k}_{m}(V, \W, \F)$, and

$(v_1, \ldots,
v_{k+m}) \in V$, $(x_1, \ldots, %x_{k}; x_{k+1}, \ldots,
x_{k+m})\in \C$,

we have

For arbitrary  $w'\in W'$, the series

\begin{eqnarray}
\label{Jnm2}
\mathcal J^k_m(\Phi)=
\sum_{q\in \C}\langle w',
E^{(m)}_{W} \Big(v_{1}, x_1;  \ldots; %\otimes \ldots\otimes
v_{m}, x_m;
P_{q}( \Phi(v_{m+1}, x_{m+1}; \ldots; %  \otimes \ldots\otimes
v_{m+k}, x_{m+k}
\Big)
\rangle,
\nn
\end{eqnarray}

is absolutely convergent when

\begin{eqnarray}
\label{granizy2}
x_{i}\ne x_{j}, \quad i\ne j, \quad
\nn
\end{eqnarray}

for $i=1, \dots, m$, and $k'=m+1, \dots, k+m$, and the sum can be analytically extended to a
rational function

in $(x_{1}, \dots,  %x_{k}; x_{k+1}, \ldots,
x_{k+m})$ with the only possible poles at

$x_{i}=x_{j}$, of orders less than or equal to

$N^k_m(v_{i}, v_{j})$, for $i, j=1, \dots, k$, $i\ne j$.

2')  For

$\Psi(v'_{1}, y_{1}; \ldots; v'_{n}, y_{n})  \in   C_{m'}^{n}(V, \W, \F)$,

$(v'_1, \ldots,
v'_{n+m'})\in V$,  and

$(y_1, \ldots,
y_{n+m'})\in \C$,

the series

\begin{eqnarray}
\label{Jnm}
&& \mathcal J^{n}_{m'}(\Psi)=
\sum_{q\in \C}\langle w', E^{(m')}_{W} \Big(v'_{1}, y_1; \ldots;
v'_{m'}, y_{m'};
\nn
P_{q}( \Psi(v'_{m'+1}, y_{m'+1}; \ldots; v'_{m'+n}, y_{m'+n})
\end{eqnarray}

is absolutely convergent when

\begin{eqnarray}
\label{granizy2}
y_{i'}\ne y_{j'}, \quad i'\ne j', \quad
\nn
\end{eqnarray}

for $i'=1, \dots, m'$, and $k''=m'+1, \dots, n +m'$, and the sum can be analytically extended to a
rational function

in $(y_{1}, \ldots,
y_{n+m'})$ with the only possible poles at

$y_{i'}=y_{j'}$, of orders less than or equal to

$N^{n}_{m'}(v'_{i'}, v'_{j'})$, for $i'$,  $j'=1, \dots, n$, $i'\ne j'$.

Thus, for the product (<ref>) we obtain

$(v''_{1}, \dots, v''_{k+n +m+m'})\in V$,

$(z_1, \ldots $ , $ z_{k+n+m+m'} )\in \C$,

we find

positive integers

$N^{k+n}_{m+m'}(v'_{i}, v'_{j})$,

depending only on $v'_{i}$ and

$v''_{j}$, for $i''$, $j''=1, \dots, k+n$, $i''\ne j''$, such that for arbitrary $w'\in W'$.

First we note

\begin{eqnarray*}
%\label{Jnm00 }
\sum_{q\in \C}\langle w', E^{(m+m')}_{W} \Big(
v''_{1}, z_{1}; \ldots;
v''_{m+m'}, z_{m+m'};
\nn
\qquad \qquad  \qquad
P_{q}\Big( \Theta(v''_{m+m'+1}, z_{m+m'+1}; \ldots; v''_{m+m'+k+n}, z_{m+m'+k+n}
\Big) \Big) \rangle
\nn
%\qquad \qquad
\sum_{l \in \Z }  \epsilon^l
\sum_{u\in V_l }
\langle w',  E^{(m)}_{W} \Big(
v_{k+1}, x_{k+1}; \ldots;
v_{k+m}, x_{k+m};
\nn
\qquad \qquad  \qquad \qquad  \qquad \qquad
P_{q} \Big(
\Phi (v_{1}, x_{1};  \ldots; v_{k}, x_{k}), \zeta_1\right)\; u \Big) \Big)\rangle
\nn
& &
\qquad   \qquad
\langle w',  E^{(m')}_{W} \Big(
v'_{n+1}, y_{n+1}; \ldots;
v'_{n+m'}, y_{n+m'};
\nn
\qquad \qquad \qquad \qquad  \qquad \qquad
P_{q}\Big(    Y^{W}_{WV}\left(
\Psi
(v'_{1}, y_{1}; \ldots; v'_{n}, y_{n}) , \zeta_{2}\right) \; \overline{u} \Big) \Big) \rangle.
\end{eqnarray*}

\begin{eqnarray*}
\sum_{l \in \Z }  \epsilon^l
\sum_{u\in V_l }
\langle w',  E^{(m+m')}_{W} \Big(
v''_{1}, z_1; \ldots;
v''_{m+m'}, z_{m+m'};
%%v_{1}\otimes \ldots\otimes v_{m};
\nn
\qquad \qquad  P_{q} \Big(
\Phi (v''_{m+m'+1}, z_{m+m'+1};  \ldots; v''_{m+m' +k}, z_{m+m'+k}), \zeta_1\right)\; u \Big) \Big)\rangle
\nn
& &
\qquad   \qquad
\langle w',  E^{(m+m')}_{W} \Big(
v''_{1}, z_1; \ldots;
v''_{m+m'}, z_{m+m'};
\nn
\qquad \qquad P_{q}\Big(    Y^{W}_{WV}\left(
\Psi
(v''_{m+m'+k+1}, z_{m+m'+k+1}; \ldots; \right.
\nn
\qquad \qquad  \qquad \qquad    \qquad \qquad \qquad \qquad \left.
v''_{m+m'+k+n}, z_{m+m'+k+n}) , \zeta_{2}\right) \; \overline{u} \Big) \Big) \rangle
\end{eqnarray*}

\begin{eqnarray*}
= \sum_{q\in \C}
\sum_{l \in \Z }  \epsilon^l
\sum_{u\in V_l }
\langle w', E^{(m+m')}_{W} \Big(
v''_{1}, z_1; \ldots; %\otimes \ldots\otimes
v''_{m+m'}, z_{m+m'};
\nn
\qquad
P_{q}\Big( e^{\zeta_1 L_W{(-1)} }\; Y_{W} %_{WV}
\left( u, -\zeta_1 )
\; \Phi (v''_{m+m'+1}, z_{m+m'+1};  \ldots; v''_{m+m'+ k }, z_{m+m'+k}) %, -\zeta_1) u%, \zeta_1
\right) %\; u
\Big) \rangle
\nn
& &
\quad   %\qquad
\langle w',  E^{(m+m')}_{W} \Big(
%%v'_{1}\otimes \ldots\otimes v'_{m'};
v''_{1}, z_1; \ldots;
v''_{m+m'}, z_{m+m'};
\nn
P_{q}\Big( e^{\zeta_2 L_W{(-1)}}\; Y_{W} %_{WV}
\left(  \overline{u}, -\zeta_2  )\;
\Psi
(v''_{m+m'+k+1}, z_{m+m'+k+1}; \ldots; \right.
\nn
\qquad \qquad     \qquad \qquad  \qquad \qquad    \qquad \qquad \qquad \qquad \left.
v''_{m+m'+k+n}, z_{m+m'+k+n})  %, -\zeta_{2}) \overline{u}
\right) %\; \overline{u}
\Big)\rangle.  %\right|
\end{eqnarray*}

The action of exponentials $e^{\zeta_a L_W{(-1)} }$, $a=1$, $2$, of the differential operator $L_W{(-1)}$,

and $W$-module vertex operators $Y_{W} \left( u, -\zeta_1 \right)$,

$Y_{W} \left( u, -\zeta_2 \right)$

shifts the grading index $q$ of $W_q$-subspaces by $\alpha \in \C$ which can be later rescaled

to $q$.

Thus, we can rewrite the last expression as

\begin{eqnarray*}
= \sum_{q\in \C}  %\sum_{u\in V }
\sum_{l \in \Z }  \epsilon^l
\sum_{u\in V_l }
\langle w', E^{(m+m')}_{W} \Big(
v''_{1}, z_1; \ldots;
v''_{m+m'}, z_{m+m'};
\nn
\qquad
e^{\zeta_1 L{_W(-1)} }\; Y_{W}
\left
( u, -\zeta_1 \big)
\;P_{q+\alpha}\Big(   \Phi(v''_{m+m'+1}, z_{m+m'+1};  \ldots; v''_{m+m'+ k }, z_{m+m'+k})
\right) %\; u
\Big) \rangle
\nn
& &
\langle w',  E^{(m+m')}_{W} \Big(
%%v'_{1}\otimes \ldots\otimes v'_{m'};
v''_{1}, z_1; \ldots;
v''_{m+m'}, z_{m+m'};
\nn
\quad   %\qquad
e^{\zeta_2 L_W{(-1)} }\; Y_{W}
\left(  \overline{u}, -\zeta_2  \Big)\;   \right.
%\qquad \qquad
\nn
\left.
\qquad P_{q+\alpha}\Big(  \Psi
(v''_{m+m'+k+1}, z_{m+m'+k+1}; \ldots; v''_{m+m'+k+n}, z_{m+m'+k+n})
\right)
\Big)\rangle
\end{eqnarray*}

\begin{eqnarray*}
= \sum_{q\in \C}
\sum_{l \in \Z }  \epsilon^l
\sum_{u\in V_l }
\langle w', E^{(m+m')}_{W} \Big(
v''_{1}, z_1; \ldots;
v''_{m+m'}, z_{m+m'};
\nn
\left(
P_{q+\alpha}\Big(   \Phi (v''_{m+m'+1}, z_{m+m'+1};  \ldots; v''_{m+m'+ k }, z_{m+m'+k}) %, -\zeta_1) , \zeta_1
\right)
\Big), \zeta_1 \Big)\; u
\rangle
\nn
\langle w',  E^{(m+m')}_{W} \Big(
%%v'_{1}\otimes \ldots\otimes v'_{m'};
v''_{1}, z_1; \ldots;
%%\otimes \ldots\otimes
v''_{m+m'}, z_{m+m'};
\nn
& &
\left(
P_{q+\alpha}\Big(  \Psi
(v''_{m+m'+k+1}, z_{m+m'+k+1}; \ldots; v''_{m+m'+k+n}, z_{m+m'+k+n})  , -\zeta_{2}) \;\overline{u}
\right)
\Big)\rangle
\end{eqnarray*}

\begin{eqnarray*}
= \sum_{q\in \C}
\sum_{\widetilde{w} \in W }
\langle w', E^{(m+m')}_{W} \Big(
v''_{1}, z_1; \ldots;
v''_{m+m'}, z_{m+m'};  \widetilde{w} \Big) \rangle
\nn
\qquad
\sum_{l \in \Z }  \epsilon^l
\sum_{u\in V_l }
\langle w',  Y^{W}_{WV}
\left(
\;P_{q+\alpha}\Big(   \Phi (v''_{m+m'+1}, z_{m+m'+1};  \ldots; v''_{m+m'+ k }, z_{m+m'+k}) , -\zeta_1)\; u
\right)
\Big)  \rangle
\nn
& &
\quad   %\qquad
\langle \widetilde{w}', E^{(m+m')}_{W} \Big(
v''_{1}, z_1; \ldots;
v''_{m+m'}, z_{m+m'};  \widetilde{w} \Big) \rangle
\nn
\langle w', Y^{W}_{WV}
\left(
P_{q+\alpha}\Big(  \Psi
(v''_{m+m'+k+1}, z_{m+m'+k+1}; \ldots; v''_{m+m'+k+n}, z_{m+m'+k+n})  , -\zeta_{2}) \;\overline{u}
\right)
\Big)\rangle
\end{eqnarray*}

\begin{eqnarray*}
= \sum_{q\in \C}
\langle w', E^{(m+m')}_{W} \Big(
v''_{1}, z_1; \ldots; %\otimes \ldots\otimes
v''_{m+m'}, z_{m+m'};  %w  \rangle
\nn
\qquad
P_{q+\alpha}\Big(   \Theta (v''_{m+m'+1}, z_{m+m'+1};  \ldots; v''_{m+m'+ k }, z_{m+m'+k}; % ) , -\zeta_1) u
\nn
& &
\qquad   \qquad
v''_{m+m'+k+1}, z_{m+m'+k+1}; \ldots; v''_{m+m'+k+n}, z_{m+m'+k+n}) \Big)
\rangle.
\end{eqnarray*}

Now note that, according to Proposition <ref>, as an element of $\W_{z_1, \ldots,  z_{k+n+m+m'}}$

\begin{eqnarray}
\label{svoloch}
&& \langle w', E^{(m+m')}_{W} \Big(
v''_{1}, z_1; \ldots; %\otimes \ldots\otimes
v''_{m+m'}, z_{m+m'};  %w  \rangle
\nn
\qquad
P_{q+\alpha}\Big(   \Theta (v''_{m+m'+1}, z_{m+m'+1};  \ldots; v''_{m+m'+ k }, z_{m+m'+k}; % ) , -\zeta_1) u
\nn
& &
\qquad   \qquad
v''_{m+m'+k+1}, z_{m+m'+k+1}; \ldots; v''_{m+m'+k+n}, z_{m+m'+k+n}) \Big)
\rangle,
\end{eqnarray}

is invariant with respect to the action of $\sigma \in S_{k+n+m+m'}$.

Thus we are able to use this invariance to show that (<ref>) is reduced to

\begin{eqnarray*}
&& \langle w', E^{(m+m')}_{W} \Big(
v''_{k+1}, z_{k+1}; \ldots; v''_{k+1+m}, z_{k+1+m};  %\otimes \ldots\otimes
v''_{n+1}, z_{n+1}; \ldots; v''_{n+1+m'}, z_{n+1+m'};   %w  \rangle
\nn
\qquad
P_{q+\alpha}\Big(   \Theta (v''_{1}, z_{1};  \ldots; v''_{k}, z_{k};
%\qquad   \qquad
v''_{k+1}, z_{k+1}; \ldots; v''_{k+n}, z_{k+n}) \Big) \Big)
\rangle
\nn
=\langle w', E^{(m+m')}_{W} \Big(
v_{k+1}, x_{k+1};  \ldots; v_{k+1+m}, x_{k+1+m};
v'_{n+1}, y_{n+1};  \ldots; v'_{n+1+m'}, y_{n+1+m'};
\nn
\qquad
P_{q+\alpha}\Big(   \Theta (v_{1}, x_{1};  \ldots; v_{k}, x_{k};
v'_{1}, y_{1}; \ldots; v'_{n}, y_{n}) \Big)
\rangle.
\end{eqnarray*}

Similarly, since

\begin{eqnarray*}
\langle w',  E^{(m)}_{W} \Big(
v''_{1}, z_1; \ldots;
v''_{m+m'}, z_{m+m'};
\nn
\qquad \qquad  P_{q} \Big(
\F (v''_{m+m'+1}, z_{m+m'+1};  \ldots; v''_{m+m' +k}, z_{m+m'+k}), \zeta_1\right)\; u \Big) \Big)\rangle,
\end{eqnarray*}

\begin{eqnarray*}
& &
%\qquad   \qquad
\langle w',  E^{(m')}_{W} \Big(
v''_{1}, z_1; \ldots;
v''_{m+m'}, z_{m+m'};
\nn
\qquad
P_{q}\Big(    Y^{W}_{WV}\left(
\F
(v''_{m+m'+k+1}, z_{m+m'+k+1}; \ldots; v''_{m+m'+k+n}, z_{m+m'+k+n}) , \zeta_{2}\right) \; \overline{u} \Big) \Big) \rangle.
\end{eqnarray*}

correspond to elements of $\W_{
z_1, \ldots,
}$ and

$\W_{z_{m+m'+k+1}, \ldots, z_{m+m'+k+n}}$,

we use Proposition <ref> again and obtain

\begin{eqnarray*}
\langle w',  E^{(m)}_{W} \Big(
v_{k+1}, x_{k+1}; \ldots;
v_{k+m}, x_{k+m};
P_{q} \Big(
\F (v_{1}, x_{1};  \ldots; v_{k}, x_{k}), \zeta_1\right)\; u \Big) \Big)\rangle
\end{eqnarray*}

\begin{eqnarray*}
& &
%\qquad   \qquad
\langle w',  E^{(m')}_{W} \Big(
v'_{n+1}, y_{n+1}; \ldots;
v'_{n+m'}, y_{n+m'};
P_{q}\Big(    Y^{W}_{WV}\left(
\F
(v'_{1}, y_{1}; \ldots; v'_{n}, y_{n}) , \zeta_{2}\right) \; \overline{u} \Big) \Big) \rangle,
\end{eqnarray*}

Thus, the assertion of Lemma follows.

Under conditions

\begin{eqnarray}
\label{granizy2}
z_{i''}\ne z_{j''}, \quad i''\ne j'', \quad
\nn
\end{eqnarray}

for $i''=1, \dots, m+m'$, and $k'''=m+m'+1, \dots, m+m'+ k+n$,

let us introduce

\begin{eqnarray}
\label{perda}
\mathcal J^{k+n}_{m+m'}(\Theta) = \sum_{q\in \C}
\langle w', E^{(m+m')}_{W} \Big(
v''_{1}, z_1; \ldots;
v''_{m+m'}, z_{m+m'};  %\right.
\nn
\left.
\qquad %\qquad
P_{q}\Big( \Theta( v''_{m+m'+1}, z_{m+m'+1}; \ldots; v''_{m+m'+k+n}, z_{m+m'+k+n})
; \epsilon \right)\Big)\rangle.
\end{eqnarray}

Using Lemma <ref> we obtain

\begin{eqnarray*}
|\mathcal J^{k+n}_{m+m'}(\Theta) |
\nn
= \left| \sum_{q\in \C}\langle w', E^{(m+m')}_{W} \Big(
v''_{1}, z_1; \ldots; %\otimes \ldots\otimes
v''_{m+m'}, z_{m+m'};  \right.
\nn
\left.
\qquad P_{q}\Big( \Theta( v''_{m+m'+1}, z_{m+m'+1}; \ldots; v''_{m+m'+k+n}, z_{m+m'+k+n})
; \epsilon)\Big)\rangle \right|
\end{eqnarray*}

\begin{eqnarray*}
= \left| \sum_{q\in \C}
\sum_{l \in \Z }  \epsilon^l
\sum_{u\in V_l }
\langle w',  E^{(m)}_{W} \Big(
v_{k+1}, x_{k+1}; \ldots;
v_{k+m}, x_{k+m};
\right.
\nn
\qquad \qquad  \qquad \qquad \qquad \qquad
P_{q} \Big(
\Phi (v_{1}, x_{1};  \ldots; v_{k}, x_{k}), \zeta_1\right)\; u \Big) \Big)\rangle
\nn
& &
\qquad   \qquad
\langle w',  E^{(m')}_{W} \Big(
v'_{n+1}, y_{n+1}; \ldots;
v'_{n+m'}, y_{n+m'};
\nn
\left.
\qquad \qquad
\qquad \qquad \qquad \qquad
P_{q}\Big(    Y^{W}_{WV}\left(
\Psi
(v'_{1}, y_{1}; \ldots; v'_{n}, y_{n}) , \zeta_{2}\right) \; \overline{u} \Big) \Big) \rangle \right|
\nn
\le \left|
\mathcal J^{k}_{m}(\F) \right| \; \left|  \mathcal J^{n}_{m'}(\F)\right|,
\end{eqnarray*}

where we have used
the invariance of (<ref>) with respect to

$\sigma \in S_{m+m'+k+n}$.

According to Definitions <ref>

$\mathcal J^{k}_{m}(\Phi)$ and $\mathcal J^{n}_{m'}(\Psi)$ in the last expression

are absolute convergent.

Thus, we infer that  $\mathcal J^{k+n}_{m+m'}(\Theta)$

is absolutely convergent, and

the sum (<ref>)

is analytically extendable to a  rational function

in $(z_{1}, \dots, z_{k+n+m+m'})$ with the only possible poles at

$x_i=x_j$, $y_{i'}=y_{j'}$, and

at $x_i=y_{j'}$, i.e., the only possible poles at

$z_{i''}=z_{j''}$, of orders less than or equal to

$N^{k+n}_{m+m'}(v''_{i''}, v''_{j''})$,

for $i''$, $j''=1, \dots, k'''$, $i''\ne j''$.

§.§ Proof of Proposition <ref>

For a vertex operator $Y_{V, W}(v,z)$ let us introduce a notation

\[
\omega_{V, W}=Y_{V, W}(v,z)\; dz^{{\rm wt} v}.
\]

Let us use notations (<ref>) and (<ref>).

According to (<ref>), the action of

$\delta_{m + m'-t}^{k + n-r}$ on $\widehat{R} \Theta(
v_1, x_1; \ldots; v_k, x_k; v'_1, y_1; \ldots; v'_k, y_n
%%v_1', z_; \ldots; v'_{k+n'-r}, z_{k+n'-r}
; \epsilon)$

is given by

\begin{eqnarray*}
\langle w',
\delta_{m + m'-t}^{k + n-r} \widehat{R} \; \Theta(
v_1, x_1; \ldots; v_k, x_k; v'_1, y_1; \ldots; v'_n, y_n
%v_1', z_; \ldots; v'_{k+n'-r}, z_{k+n'-r}
; \epsilon) \rangle
\nn
&& \quad =
\langle w', \sum_{i=1}^{k%+n
}(-1)^{i} \;
\widehat{R} \; \Theta ( \widetilde{v}_1, z_1; \ldots;  \widetilde{v}_{i-1}, z_{i-1}; \;  \omega_V (\widetilde{v}_i, z_i
- z_{i+1})
\widetilde{v}_{i+1}, z_{i+1}; \; \widetilde{v}_{i+2}, z_{i+2};
\nn
&& \qquad \qquad \qquad
\ldots;  \widetilde{v}_k, z_k;  \widetilde{v}_{k+1}, z_{k+1}; \ldots; \widetilde{v}_{k+n}, z_{k+n}; \epsilon ) \rangle
\end{eqnarray*}

\begin{eqnarray*}
&& \qquad +
\sum_{i=1}^{n-r}(-1)^{i} \; \langle w',
\Theta \left( \widetilde{v}_1, z_1; \ldots; \widetilde{v}_k, z_k;
\widetilde{v}_{k+1}, z_{k+1}; \ldots; \widetilde{v}_{k+i-1}, z_{k+i-1};
\right.
\nn
&&\qquad \qquad \qquad
\omega_V \left(\widetilde{v}_{k+i}, z_{k+i}
- z_{k+i+1} ) \;
\widetilde{v}_{k+i+1}, z_{k+i+1}; \right.
\nn
\left.
\qquad \qquad \qquad \qquad \qquad \qquad \widetilde{v}_{k+i+2}, z_{k+i+2}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r}; \epsilon \right) \rangle
\end{eqnarray*}

\begin{eqnarray*}
&& \qquad +  \langle w',
\omega_W \left(\widetilde{v}_1, z_1
\right) \; \Theta (\widetilde{v}_2, z_2
; \ldots; \widetilde{v}_{k}, z_k; \widetilde{v}_{k+1}, z_{k+1}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r}; \epsilon
)  \rangle
\end{eqnarray*}

\begin{eqnarray*}
& &\qquad +  \langle w, (-1)^{k+n+1-r}
\omega_W(\widetilde{v}_{k+n-r+1}, z_{k+n-r+1}
\;
\nn
\qquad \qquad \qquad \F(\widetilde{v}_1, z_1
; \ldots; \widetilde{v}_k, z_k; \widetilde{v}_{k+1}, z_{k+1}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r}; \epsilon ) \rangle
\end{eqnarray*}

\begin{eqnarray*}
&& \quad =
\sum_{l \in \Z }  \epsilon^l
\langle w', \sum_{i=1}^{k}(-1)^{i} \; Y^W_{VW}(
\Theta ( \widetilde{v}_1, z_1; \ldots;  \widetilde{v}_{i-1}, z_{i-1}; \; \omega_V (\widetilde{v}_i, z_i
- z_{i+1})
\widetilde{v}_{i+1}, z_{i+1}; \;
\nn
&& \qquad \qquad \qquad
\widetilde{v}_{i+2}, z_{i+2};  \ldots;  \widetilde{v}_k, z_k), \zeta_1) u \rangle
\nn
\qquad \qquad \qquad \qquad \qquad \qquad
\langle w', Y^W_{VW}( \Theta(\widetilde{v}_{k+1}, z_{k+1}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r}), \zeta_2)
\overline{u} \rangle
\end{eqnarray*}

\begin{eqnarray*}
&& \qquad +
\sum_{l \in \Z }  \epsilon^l  \sum_{i=1}^{n-r}(-1)^{i} \; \langle w',  Y^W_{VW}(
\Phi \left( \widetilde{v}_1, z_1; \ldots; \widetilde{v}_k, z_k), \zeta_1\right) u \rangle
\nn
\qquad \qquad \qquad
\langle w',
Y^W_{VW}( \Psi(\widetilde{v}_{k+1}, z_{k+1}; \ldots; \widetilde{v}_{k+i-1}, z_{k+i-1};
\nn
\qquad \qquad \qquad  \qquad \omega_V \left(\widetilde{v}_i, z_{k+i}
- z_{k+i+1}\right) \;
\widetilde{v}_{k+i+1}, z_{k+i+1}; \widetilde{v}_{k+i+2}, z_{k+i+2};
\nn
\qquad \qquad \qquad  \qquad \qquad  \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r} ), \zeta_2) \overline{u} \rangle
\end{eqnarray*}

\begin{eqnarray*}
&& \qquad +  \sum_{l \in \Z }  \epsilon^l \langle w', Y^W_{VW}(
\omega_W \left(\widetilde{v}_1, z_1 \right) \; \Phi (\widetilde{v}_2, z_2 ; \ldots; \widetilde{v}_{k}, z_k), \zeta_1) u \rangle
\nn
\qquad \qquad \langle w', Y^W_{VW}( \Psi( \widetilde{v}_{k+1}, z_{k+1}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r} ), \zeta_2) \overline{u} \rangle
\end{eqnarray*}

\begin{eqnarray*}
\qquad +  \sum_{l \in \Z }  \epsilon^l  \langle w', Y^W_{VW}( (-1)^{k+1}
\omega_W \left(\widetilde{v}_{k+1}, z_{k+1}
\right) \;  \Phi (\widetilde{v}_1, z_1 ; \ldots; \widetilde{v}_{k}, z_k), \zeta_1) u\rangle
\nn
&& \qquad \qquad \qquad
\langle w', Y^W_{VW}(
\Psi( \widetilde{v}_{k+2}, z_{k+2}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r}), \zeta_2) \overline{u} \rangle
\end{eqnarray*}

\begin{eqnarray*}
\qquad -
\sum_{l \in \Z }  \epsilon^l \langle w', (-1)^{k+1}   \langle w',  Y^W_{VW}(
\omega_W \left(\widetilde{v}_{k+1}, z_{k+1}
\right) \; \Phi (\widetilde{v}_1, z_1 ; \ldots; \widetilde{v}_{k}, z_k), \zeta_1) u \rangle
\nn
\qquad \qquad  \qquad
\langle w',  Y^W_{VW}( \Psi(\widetilde{v}_{k+2}, z_{k+2}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r}),  \zeta_2)  \overline{u}
\rangle
\nn
\qquad +  \sum_{l \in \Z }  \epsilon^l  \langle w', Y^W_{VW}(
\Phi(\widetilde{v}_1, z_1; \ldots; \widetilde{v}_k, z_k), \zeta_1)  u\rangle
\nn
\qquad \qquad
\langle w', Y^W_{VW}(
\omega_W(\widetilde{v}_{k+n-r+1}, z_{k+n-r+1})\;
\nn
\qquad \qquad \qquad  \qquad  \Psi(\widetilde{v}_{k+1}, z_{k+1}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r}  ), \zeta_2)  \overline{u}\rangle
\nn
&& \qquad
- \sum_{l \in \Z }  \epsilon^l \langle w', Y^W_{VW}(
\Phi(\widetilde{v}_1, z_1; \ldots; \widetilde{v}_k, z_k), \zeta_1) \rangle
\nn
\qquad \qquad \langle w',  Y^W_{VW}(
\omega_W(\widetilde{v}_{k+n-r+1}, z_{k+n-r+1})
\nn
\qquad \qquad \qquad  \qquad  \Psi( \widetilde{v}_{k+1}, z_{k+1}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r}  ), \zeta_2) \rangle
\end{eqnarray*}

\begin{eqnarray*}
&& \quad =
\sum_{l \in \Z }  \epsilon^l
\langle w',  \; Y^W_{VW}(
\delta^k_m\Phi ( \widetilde{v}_1, z_1; \ldots;  \widetilde{v}_k, z_k), \zeta_1) u \rangle
\nn
\qquad \qquad \qquad \qquad \langle w', Y^W_{VW}( \Psi(\widetilde{v}_{k+1}, z_{k+1}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r}), \zeta_2)
\overline{u} \rangle
\nn
&& \qquad + (-1)^k
\sum_{l \in \Z }  \epsilon^l   \langle w',  Y^W_{VW}(
\Phi \left( \widetilde{v}_1, z_1; \ldots; \widetilde{v}_k, z_k), \zeta_1 \right) u \rangle
\nn
\qquad \qquad \qquad
\langle w',  Y^W_{VW}( \delta^{n-r}_{m'-t}  \Psi(\widetilde{v}_{k+1}, z_{k+1}; \ldots;  \widetilde{v}_{k+n-r}, z_{k+n-r} ), \zeta_2 ) \overline{u} \rangle
\end{eqnarray*}

\begin{eqnarray*}
&& \quad =
\langle w',
\delta^k_m\Phi ( \widetilde{v}_1, z_1; \ldots;  \widetilde{v}_k, z_k) \cdot
%\qquad \qquad \qquad \qquad
\langle w',  \Psi(\widetilde{v}_{k+1}, z_{k+1}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r}) \rangle
\nn
&& \qquad + (-1)^k
\langle w',
\Phi \left( \widetilde{v}_1, z_1; \ldots; \widetilde{v}_k, z_k\right) \cdot_{\epsilon}
\delta^{n-r}_{m'-t}  \Psi(\widetilde{v}_{k+1}, z_{k+1}; \ldots;  \widetilde{v}_{k+n-r}, z_{k+n-r} ) \rangle,
\end{eqnarray*}

\begin{eqnarray*}
&& \sum_{l \in \Z }  \epsilon^l \langle w', (-1)^{k+1}   Y^W_{VW}(
\omega_W \left(\widetilde{v}_{k+1}, z_{k+1}
\right) \; \Phi (\widetilde{v}_1, z_1 ; \ldots; \widetilde{v}_{k}, z_k), \zeta_1) u \rangle
\nn
\qquad \qquad \langle w',  Y^W_{VW}( \Psi(\widetilde{v}_{k+2}, z_{k+2}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r}),  \zeta_2) \overline{u}
\rangle
\end{eqnarray*}

\begin{eqnarray*}
=\sum_{l \in \Z }  \epsilon^l \langle w', (-1)^{k+1} e^{\zeta_1 L_W{(-1)}}   Y_W(u, -\zeta_1)  \;
\omega_W \left(\widetilde{v}_{k+1}, z_{k+1}
\right) \; \Phi (\widetilde{v}_1, z_1 ; \ldots; \widetilde{v}_{k}, z_k) \rangle
\nn
\qquad \qquad \langle w',  Y^W_{VW}( \Psi(\widetilde{v}_{k+2}, z_{k+2}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r}),  \zeta_2) \overline{u}
\rangle
\end{eqnarray*}

\begin{eqnarray*}
=\sum_{l \in \Z }  \epsilon^l \langle w', (-1)^{k+1} e^{\zeta_1 L_W{(-1)}}  \omega_W \left(\widetilde{v}_{k+1}, z_{k+1}  \right)
Y_W(u, -\zeta_1)  \;
\; \Phi (\widetilde{v}_1, z_1 ; \ldots; \widetilde{v}_{k}, z_k) \rangle
\nn
\qquad \qquad \langle w',  Y^W_{VW}( \Psi(\widetilde{v}_{k+2}, z_{k+2}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r}),  \zeta_2) \overline{u}
\rangle
\end{eqnarray*}

\begin{eqnarray*}
\nn
= \sum\limits_{v\in V}
%\sum_{l \in \Z }  \epsilon^l
\langle w', (-1)^{k+1} \;
\omega_W \left(\widetilde{v}_{k+1}, z_{k+1} +\zeta_1  \right)\;  e^{\zeta_1 L_W{(-1)}}
Y_W(u, -\zeta_1)  \;
\; \Phi (\widetilde{v}_1, z_1 ; \ldots; \widetilde{v}_{k}, z_k) \rangle
\nn
\qquad \qquad \langle w',  Y^W_{VW}( \Psi(\widetilde{v}_{k+2}, z_{k+2}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r}),  \zeta_2) \overline{u}
\rangle
\end{eqnarray*}

\begin{eqnarray*}
=\sum\limits_{v\in V}
\sum_{u\in V_l }
\sum_{l \in \Z }  \epsilon^l
\sum_{u\in V_l }
\langle v', (-1)^{k+1} \; \omega_W \left(\widetilde{v}_{k+1}, z_{k+1}+\zeta_1  \right) w \rangle
\nn
\qquad \qquad \langle w',  e^{\zeta_1 L_W{(-1)}}
Y_W(u, -\zeta_1)  \;
\; \Phi (\widetilde{v}_1, z_1 ; \ldots; \widetilde{v}_{k}, z_k) \rangle
\nn
\qquad \qquad \langle w',  Y^W_{VW}( \Psi(\widetilde{v}_{k+2}, z_{k+2}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r}),  \zeta_2) \overline{u}
\rangle
\end{eqnarray*}

\begin{eqnarray*}
\sum_{l \in \Z }  \epsilon^l
%\qqaud \qquad
\langle w',  e^{\zeta_1 L_W{(-1)}}
Y_W(u, -\zeta_1)  \;
\; \Phi (\widetilde{v}_1, z_1 ; \ldots; \widetilde{v}_{k}, z_k) \rangle
\nn
\qquad \qquad \sum\limits_{v\in V} \langle v', (-1)^{k+1} \; \omega_W \left(\widetilde{v}_{k+1}, z_{k+1}+\zeta_1  \right) w \rangle
\nn
\langle w',  Y^W_{VW}( \Psi(\widetilde{v}_{k+2}, z_{k+2}; \ldots;
\nn
\qquad \qquad \qquad \qquad \widetilde{v}_{k+n-r}, z_{k+n-r}),  \zeta_2) \overline{u}
\rangle
\end{eqnarray*}

\begin{eqnarray*}
\sum_{l \in \Z }  \epsilon^l
%\qqaud \qquad
\langle w',
\Phi (\widetilde{v}_1, z_1 ; \ldots; \widetilde{v}_{k}, z_k) , \zeta_1) u \; \rangle
\nn
\qquad \qquad
\langle w', (-1)^{k+1} \; \omega_W \left(\widetilde{v}_{k+1}, z_{k+1}+\zeta_1  \right)
\;
\nn
\qquad \qquad  Y^W_{VW}( \Psi(\widetilde{v}_{k+2}, z_{k+2}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r}),  \zeta_2) \overline{u}
\rangle
\end{eqnarray*}

\begin{eqnarray*}
\sum_{l \in \Z }  \epsilon^l
%\qqaud \qquad
\langle w',
\Phi (\widetilde{v}_1, z_1 ; \ldots; \widetilde{v}_{k}, z_k) , \zeta_1) u \; \rangle
\nn
\qquad \qquad
\langle w', (-1)^{k+1} \; \omega_W \left(\widetilde{v}_{k+1}, z_{k+1}+\zeta_1  \right)
\nn
\qquad \qquad
\; e^{\zeta_2 L_W{(-1)}}    Y_W(\overline{u}, -\zeta_2) \; \Psi(\widetilde{v}_{k+2}, z_{k+2}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r})
\rangle
\end{eqnarray*}

\begin{eqnarray*}
\sum_{l \in \Z }  \epsilon^l
%\qqaud \qquad
\langle w',
\Phi (\widetilde{v}_1, z_1 ; \ldots; \widetilde{v}_{k}, z_k) , \zeta_1) u \; \rangle
\nn
\qquad
\langle w', (-1)^{k+1} \;
\; e^{\zeta_2 L_W{(-1)}} \;   Y_W(\overline{u}, -\zeta_2)
\;
\omega_W \left(\widetilde{v}_{k+1}, z_{k+1}+\zeta_1-\zeta_2 \right)
\nn
\qquad \qquad \; \Psi(\widetilde{v}_{k+2}, z_{k+2}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r})
\rangle
\end{eqnarray*}

\begin{eqnarray*}
\qquad \qquad
= \sum_{l \in \Z }  \epsilon^l \langle w', Y^W_{VW}(
\Phi(\widetilde{v}_1, z_1; \ldots; \widetilde{v}_k, z_k), \zeta_1) u \rangle
\nn
\qquad \qquad \langle w',  Y^W_{VW}(
\omega_W(\widetilde{v}_{k+1}, z_{k+1}) \; \Psi( \widetilde{v}_{k+2}, z_{k+2}; \ldots; \widetilde{v}_{k+n-r}, z_{k+n-r}  ), \zeta_2) \overline{u} \rangle,
\end{eqnarray*}

due to locality (<ref>) of vertex opertors, and arbitrarness of $\widetilde{v}_{k+1}\in V$ and $z_{k+1}$,

we can always put

\[
\omega_W \left(\widetilde{v}_{k+1}, z_{k+1}+\zeta_1-\zeta_2 \right)  =\omega_W(\widetilde{v}_{k+2}, z_{k+2}),
\]

for $\widetilde{v}_{k+1}=\widetilde{v}_{k+2}$, $z_{k+2}= z_{k+1}+\zeta_2-\zeta_1$.

The statement of the proposition for $\delta^2_{ex}$ (<ref>) can be checked accordingly.

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