i_\mu })^t.
    \label{eq:maid1}
  \end{aligned}$$ Then $$\begin{aligned}
    E_i F_{\beta _{n+1}}^{{b^{\chi}} (\beta _{n+1})-1}
    F_{\beta _{n+2}}^{{b^{\chi}} (\beta _{n+2})-1}\cdots
    F_{\beta _{n+\mu -1}}^{{b^{\chi}} (\beta _{n+\mu -1})-1} F_{\beta _\mu }^t \times
    \qquad &
    \\
    F_{\beta _{\mu -1}}^{{b^{\chi}} (\beta _{\mu -1})-1}\cdots
    F_{\beta _2}^{{b^{\chi}} (\beta _2)-1}
    F_{\beta _1}^{{b^{\chi}} (\beta _1)-1}v_\Lambda =0&
  \end{aligned}$$ for all $i\in I$.

By Eq.  it suffices to prove that $$\begin{aligned}
    F_{\beta _{n+\nu }} F_{\beta _{n+1}}^{{b^{\chi}} (\beta _{n+1})-1}
    F_{\beta _{n+2}}^{{b^{\chi}} (\beta _{n+2})-1}\cdots
    F_{\beta _{n+\mu -1}}^{{b^{\chi}} (\beta _{n+\mu -1})-1} F_{\beta _\mu }^t \times
    \qquad &
    \\
    F_{\beta _{\mu -1}}^{{b^{\chi}} (\beta _{\mu -1})-1}\cdots
    F_{\beta _2}^{{b^{\chi}} (\beta _2)-1}
    F_{\beta _1}^{{b^{\chi}} (\beta _1)-1}v_\Lambda =0&
  \end{aligned}
    \label{eq:hwv}$$ for all $\nu \in \{1,2,\dots ,n\}$. Let first $\nu \in \{1,2,\dots ,\mu -1\}$. Then $$F_{\beta _{n+\nu }} F_{\beta _{n+1}}^{{b^{\chi}} (\beta _{n+1})-1}
  F_{\beta _{n+2}}^{{b^{\chi}} (\beta _{n+2})-1}\cdots
  F_{\beta _{n+\mu -1}}^{{b^{\chi}} (\beta _{n+\mu -1})-1} =0$$ by Thms. \[th:PBW\] and \[th:EErel\] by considering the ${\mathbb{Z}}^I$-degree. Thus Eq.  holds. Let now $\nu \in \{\mu ,\mu +1,\dots ,n\}$, $\Lambda _\mu ={t}_{i_{\mu -1}}\cdots {t}_{i_2}{t}_{i_1}^\chi (\Lambda )$, and $\beta '_\kappa =1_{\chi _\mu }{\sigma }_{i_\mu }{\sigma }_{i_{\mu +1}}\cdots
  {\sigma }_{i_{\mu +\kappa -2}}({\alpha }_{i_{\mu +\kappa -1}})$ for all $\kappa \in \{1,2,\dots ,2n\}$. Then by Lemma \[le:VTinv\], Eq.  is equivalent to $${\rho ^{\chi _\mu }}({\alpha }_{i_\mu })\Lambda _\mu (K_{i_\mu }L_{i_\mu }^{-1})
    ={\rho ^{\chi _\mu }}({\alpha }_{i_\mu })^t,$$ and ${\alpha }_{i_\mu }=\beta '_1$. Thus we can apply Lemma \[le:FFv\] (with $\Lambda _\mu $ instead of $\Lambda $, $\chi _\mu $ instead of $\chi $, $\beta '_\kappa $ instead of $\beta _\kappa $, and $\mu 