&=H^{(L)}(\lambda+\mu+\partial)(z_3,z_1)
i_{z_1}(z_1-z_2-\lambda-\partial)^{-1}
L(z_2)
\label{eq:a''}\\
&-H^{(L)}(\lambda+\mu+\partial)(z_3,z_2)
i_{z_1}(z_2-z_1-\mu-\partial)^{-1}
L(z_1)
\label{eq:b''}\\
&-H^{(L)}(\lambda+\mu+\partial)(z_3,z_2+\lambda+\partial)
i_{z_1}(z_1-z_2-\lambda-\partial)^{-1}
L^*(-z_1+\lambda)
\label{eq:c''}\\
&+H^{(L)}(\lambda+\mu+\partial)(z_3,z_1+\mu+\partial)
i_{z_1}(z_2-z_1-\mu-\partial)^{-1}
L^*(-z_2+\mu)
\label{eq:d''}\,.\end{aligned}$$ By Lemma \[20130925:lem1\](b) we have the following identity $$\eqref{eq:a}+\eqref{eq:c}-\eqref{eq:b'}-\eqref{eq:d'}
=\eqref{eq:a''}+\eqref{eq:c''}\,.$$ Hence, equation is proved once we show that $$\label{20130925:eq_mid}
\eqref{eq:b}+\eqref{eq:d}-\eqref{eq:a'}-\eqref{eq:c'}
=\eqref{eq:b''}+\eqref{eq:d''}\,.$$ Using equation with $\lambda$ and $\mu$ exchanged, and with $z_1$ and $z_2$ exchanged, we get that equation is equivalent to the following equation $$\begin{array}{l}
\vphantom{\Big(}
H^{(L)}(\lambda+\mu+\partial)(z_3,z_2)
\delta(z_2-z_1-\mu-\partial)
L(z_1)
\\
\vphantom{\Big(}
-H^{(L)}(\lambda+\mu+\partial)(z_3,z_1+\mu+\partial)
\delta(z_2-z_1-\mu-\partial)
L^*(-z_2+\mu)=0\,,
\end{array}$$ which holds by the properties of the $\delta$-function.

\[subalgebra\] Let ${\mathcal{V}}$ be a differential algebra, endowed with a $\lambda$-bracket $\{\cdot\,_\lambda\,\cdot\}$. Let $L(\partial)\in{\mathcal{V}}((\partial^{-1}))$ be a pseudodifferential operator, and let ${\mathcal{U}}\subset{\mathcal{V}}$ be the differential subalgebra generated by the coefficients of $L(\partial)$. Clearly, if $L(\partial)$ is of Adler type, then $\{\cdot\,_\lambda\,\cdot\}$ restricts to a $\lambda$-bracket of ${\mathcal{U}}$. Lemma \[20131027:prop1\], together with Theorem \[master\], is saying that ${\mathcal{U}}$, with this $\lambda$-bracket, is a PVA.

The generic pseudodifferential operator of order N and the corresponding AGD bi-PVA V\_N {#sec:2.2b}
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Let $N$ be a positive integer. Consider the algebra of differential polynomials ${\mathcal{V}}_N^\infty={\mathbb{F}}[u_i^{(n)}\mid i\in I,n\in{\mathbb{Z}}_+]$, where, as before, $I=\{-N,-N+1,-N+2,\dots\}$. The *generic* pseudodifferential operator on ${\mathcal{V}}_N^\infty$ is, by definition, $$\label{lcap}
L(\partial)
=\partial^N+u_{-N}\partial^{N-1}+u_{-N+1}\partial^{-N-2}+\ldots
=\sum_{n\leq N}u_{-n-1}\partial^n\in{\mathcal{V}}_N^\infty((\partial^{-1})),$$ where $u_{-N-1}=1$. For $c\in{\mathbb{F}}$, let $H^{(L-c)}(\partial)(z,w)\in{\mathcal{V}}_N^\infty[\partial]((z^{-1},w^{-1}))$ be the corresponding series, as 