By Eq. 1.2.4 we have $u_{i_{\infty}}=-u_{i_{0}}-\sum_{m\in L_{\sigma\sigma^{\prime}}}(C\cdot D_{m})u_{m}$, so

	$\displaystyle\textrm{for $i\in L_{\sigma\sigma^{\prime}}$, }\quad W^{\sigma^{\prime}}_{i}$	$\displaystyle=-\frac{\det(\Delta,(u_{l})_{l\in L_{\sigma\sigma^{\prime}}\setminus\{i\}},u_{i_{\infty}})}{\det(u_{i},(u_{l})_{l\in L_{\sigma\sigma^{\prime}}\setminus\{i\}},u_{i_{\infty}})}$
		$\displaystyle=-\frac{\det(\Delta,(u_{l})_{l\in L_{\sigma\sigma^{\prime}}\setminus\{i\}},u_{i_{0}})}{\det(u_{i},(u_{l})_{l\in L_{\sigma\sigma^{\prime}}\setminus\{i\}},u_{i_{0}})}-\sum_{m\in L_{\sigma\sigma^{\prime}}}(C\cdot D_{m})\frac{\det(\Delta,(u_{l})_{l\in L_{\sigma\sigma^{\prime}}\setminus\{i\}},u_{m})}{\det(u_{i},(u_{l})_{l\in L_{\sigma\sigma^{\prime}}\setminus\{i\}},u_{i_{0}})}$
		$\displaystyle=W^{\sigma}_{i}-(C\cdot D_{i})\frac{\det(\Delta,(u_{l})_{l\in L_{\sigma\sigma^{\prime}}\setminus\{i\}},u_{i})}{\det(u_{i},(u_{l})_{l\in L_{\sigma\sigma^{\prime}}\setminus\{i\}},u_{i_{0}})}=W^{\sigma}_{i}-(C\cdot D_{i})W^{\sigma}_{i_{0}};$
	$\displaystyle\textrm{for $i=i_{\infty}$, }\quad W^{\sigma^{\prime}}_{i_{\infty}}$	$\displaystyle=\frac{\det(\Delta,(u_{l})_{l\in L_{\sigma\sigma^{\prime}}})}{\det(u_{i_{\infty}},(u_{l})_{l\in L_{\sigma\sigma^{\prime}}})}=-\frac{\det(\Delta,(u_{l})_{l\in L_{\sigma\sigma^{\prime}}})}{\det(u_{i_{0}},(u_{l})_{l\in L_{\sigma\sigma^{\prime}}})}=-W^{\sigma}_{i_{0}}.$

For $j\in J$

	$\displaystyle\sum_{i\in L_{\sigma^{\prime}}}\lambda_{j,i}W^{\sigma^{\prime}}_{i}$	$\displaystyle=-\lambda_{j,i_{\infty}}W^{\sigma}_{i_{0}}+\sum_{i\in L_{\sigma\sigma^{\prime}}}\lambda_{j,i}\left(W^{\sigma}_{i}-(C\cdot D_{i})W^{\sigma}_{i_{0}}\right)$
		$\displaystyle=\Big(-\lambda_{j,i_{\infty}}-\sum_{i\in L_{\sigma\sigma^{\prime}}}\lambda_{j,i}(C\cdot D_{i})-\lambda_{j,i_{0}}\Big)W^{\sigma}_{i_{0}}+\sum_{i\in L_{\sigma}}\lambda_{j,i}W^{\sigma}_{i}$
		$\displaystyle=(C\cdot D_{j})W^{\sigma}_{i_{0}}+\sum_{i\in L_{\sigma}}\lambda_{i,j}W^{\sigma}_{i}\hskip 30.0pt\textrm{by \lx@cref{creftype~refnum}{equ fan}}$
	$\displaystyle W^{\sigma^{\prime}}_{j}$	$\displaystyle=-D_{j}-\sum_{l\in L}\lambda_{j,l}D_{l}-\sum_{i\in L_{\sigma^{\prime}}}\lambda_{j,i}W^{\sigma^{\prime}}_{i}$
		$\displaystyle=-D_{j}-\sum_{l\in L}\lambda_{j,l}D_{l}-\sum_{i\in L_{\sigma}}\lambda_{j,i}W^{\sigma}_{i}-(C\cdot D_{j})W^{\sigma}_{i_{0}}=W^{\sigma}_{j}-(C\cdot D_{j})W^{\sigma}_{i_{0}}.$

These relations can be summed up as $\left(\begin{matrix}W^{\sigma^{\prime}}_{i}\\ W^{\sigma^{\prime}}_{i_{\infty}}\\ W^{\sigma^{\prime}}_{j}\end{matrix}\right)=M_{\mathcal{B}^{\prime}\mathcal{B}}\left(\begin{matrix}W^{\sigma}_{i}\\ W^{\sigma}_{i_{0}}\\ W^{\sigma}_{j}\end{matrix}\right)$, i.e. $W^{\sigma^{\prime}}=(M_{\mathcal{B}^{\prime}\mathcal{B}}\otimes\Id)W^{\sigma}$. ∎