0P6Q Proof. Let A=τ∘ςi′′A=\tau\circ\varsigma^{\prime\prime}_{i} and B=ςi′′∘E2iF1iτB=\varsigma^{\prime\prime}_{i}\circ E_{2}^{i}F_{1}^{i}\tau. We have a21=a22=a31=a32=a33=a34=a41=a42=a43=0a_{21}=a_{22}=a_{31}=a_{32}=a_{33}=a_{34}=a_{41}=a_{42}=a_{43}=0 a11\displaystyle a_{11} =λ(12)∘E22ςi∘λ(2⋯2i+2)∘λ(1⋯2i+1)\displaystyle=\lambda_{(12)}\circ E_{2}^{2}\varsigma_{i}\circ\lambda_{(2\cdots 2i+2)}\circ\lambda_{(1\cdots 2i+1)} =E22ςi∘λ(1⋯2i+2)∘λ(1⋯2i+1)\displaystyle=E_{2}^{2}\varsigma_{i}\circ\lambda_{(1\cdots 2i+2)}\circ\lambda_{(1\cdots 2i+1)} a12\displaystyle a_{12} =∑r=1iλ(12)∘E22ςi−1∘E2i+1F1i−1ε1∘λ(2⋯r+1)∘λ(2i+1⋯i+r+1)∘λ(1⋯2i+1)\displaystyle=\sum_{r=1}^{i}\lambda_{(12)}\circ E_{2}^{2}\varsigma_{i-1}\circ E_{2}^{i+1}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(2\cdots r+1)}\circ\lambda_{(2i+1\cdots i+r+1)}\circ\lambda_{(1\cdots 2i+1)} =∑r=1iE22ςi−1∘E2i+1F1i−1ε1∘λ(12)∘λ(1⋯i+1)∘λ(1⋯r)∘λ(2i+1⋯i+r+1)∘λ(i+1⋯2i+1)\displaystyle=\sum_{r=1}^{i}E_{2}^{2}\varsigma_{i-1}\circ E_{2}^{i+1}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(12)}\circ\lambda_{(1\cdots i+1)}\circ\lambda_{(1\cdots r)}\circ\lambda_{(2i+1\cdots i+r+1)}\circ\lambda_{(i+1\cdots 2i+1)} =0\displaystyle=0 a13\displaystyle a_{13} =∑r=1iλ(12)∘E22ςi−1∘λ(2⋯2i)∘E2iF1i−1ε1E2∘λ(1⋯r)(2i⋯i+r)\displaystyle=\sum_{r=1}^{i}\lambda_{(12)}\circ E_{2}^{2}\varsigma_{i-1}\circ\lambda_{(2\cdots 2i)}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}E_{2}\circ\lambda_{(1\cdots r)(2i\cdots i+r)} =∑r=1iE22ςi−1∘λ(1⋯2i)∘E2iF1i−1ε1E2∘λ(1⋯r)(2i⋯i+r)∘λ(2i+1,2i+2)∘E2iF1i−1σ−1\displaystyle=\sum_{r=1}^{i}E_{2}^{2}\varsigma_{i-1}\circ\lambda_{(1\cdots 2i)}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}E_{2}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}\circ\lambda_{(2i+1,2i+2)}\circ E_{2}^{i}F_{1}^{i-1}\sigma^{-1} =∑r=1iE22ςi−1∘λ(1⋯2i)∘E2iF1i−1E2ε1∘λ(2i,2i+1)∘λ(2i⋯i+r)∘λ(1⋯r)∘E2iF1i−1σ−1\displaystyle=\sum_{r=1}^{i}E_{2}^{2}\varsigma_{i-1}\circ\lambda_{(1\cdots 2i)}\circ E_{2}^{i}F_{1}^{i-1}E_{2}\varepsilon_{1}\circ\lambda_{(2i,2i+1)}\circ\lambda_{(2i\cdots i+r)}\circ\lambda_{(1\cdots r)}\circ E_{2}^{i}F_{1}^{i-1}\sigma^{-1} =∑r=1iE22ςi−1∘E2i+1F1i−1ε1∘λ(1⋯2i+1)∘λ(2i⋯i+r)∘λ(1⋯r)∘E2iF1i−1σ−1\displaystyle=\sum_{r=1}^{i}E_{2}^{2}\varsigma_{i-1}\circ E_{2}^{i+1}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(1\cdots 2i+1)}\circ\lambda_{(2i\cdots i+r)}\circ\lambda_{(1\cdots r)}\circ E_{2}^{i}F_{1}^{i-1}\sigma^{-1} a14\displaystyle a_{14} =∑1≤r≤i1≤s<iλ(12)∘E22ςi−2∘E2iF1i−2ε1∘λ(2⋯s+1)(2i−1⋯i+s)∘E2iF1i−1ε1E1∘λ(1⋯r)(2i⋯i+r)\displaystyle=\sum_{\begin{subarray}{c}1\leq r\leq i\\ 1\leq s<i\end{subarray}}\lambda_{(12)}\circ E_{2}^{2}\varsigma_{i-2}\circ E_{2}^{i}F_{1}^{i-2}\varepsilon_{1}\circ\lambda_{(2\cdots s+1)(2i-1\cdots i+s)}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}E_{1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)} =∑1≤r≤i1≤s<iE22ςi−2∘E2iF1i−2ε1∘E2iF1i−1ε1E1∘λ(1⋯s+1)∘λ(1⋯r)∘λ(2i−1⋯i+s)∘λ(2i⋯i+r)\displaystyle=\sum_{\begin{subarray}{c}1\leq r\leq i\\ 1\leq s<i\end{subarray}}E_{2}^{2}\varsigma_{i-2}\circ E_{2}^{i}F_{1}^{i-2}\varepsilon_{1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}E_{1}\circ\lambda_{(1\cdots s+1)}\circ\lambda_{(1\cdots r)}\circ\lambda_{(2i-1\cdots i+s)}\circ\lambda_{(2i\cdots i+r)} =∑1≤r≤s<iE22ςi−2∘E2iF1i−2ε1∘E2iF1i−1ε1E1∘λ(2⋯r+1)∘λ(1⋯s+1)∘λ(2i⋯i+r)∘λ(2i⋯i+s+1)\displaystyle=\sum_{1\leq r\leq s<i}E_{2}^{2}\varsigma_{i-2}\circ E_{2}^{i}F_{1}^{i-2}\varepsilon_{1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}E_{1}\circ\lambda_{(2\cdots r+1)}\circ\lambda_{(1\cdots s+1)}\circ\lambda_{(2i\cdots i+r)}\circ\lambda_{(2i\cdots i+s+1)} a23\displaystyle a_{23} =σ−1∘E1E2ςi∘λ(2⋯2i+2)∘λ(1⋯2i+1)\displaystyle=\sigma^{-1}\circ E_{1}E_{2}\varsigma_{i}\circ\lambda_{(2\cdots 2i+2)}\circ\lambda_{(1\cdots 2i+1)} =σ−1∘E1E2ςi∘λ(2⋯2i+2)∘λ(1⋯2i+1)∘λ(2i+1,2i+2)∘E2iF1iσ−1\displaystyle=\sigma^{-1}\circ E_{1}E_{2}\varsigma_{i}\circ\lambda_{(2\cdots 2i+2)}\circ\lambda_{(1\cdots 2i+1)}\circ\lambda_{(2i+1,2i+2)}\circ E_{2}^{i}F_{1}^{i}\sigma^{-1} =σ−1∘E1E2ςi∘λ(12)∘λ(2⋯2i+2)∘λ(1⋯2i+1)∘E2iF1iσ−1\displaystyle=\sigma^{-1}\circ E_{1}E_{2}\varsigma_{i}\circ\lambda_{(12)}\circ\lambda_{(2\cdots 2i+2)}\circ\lambda_{(1\cdots 2i+1)}\circ E_{2}^{i}F_{1}^{i}\sigma^{-1} =E1E2ςi∘λ(2⋯2i+2)∘λ(1⋯2i+1)∘E2iF1iσ−1\displaystyle=E_{1}E_{2}\varsigma_{i}\circ\lambda_{(2\cdots 2i+2)}\circ\lambda_{(1\cdots 2i+1)}\circ E_{2}^{i}F_{1}^{i}\sigma^{-1} a24\displaystyle a_{24} =∑r=1iσ−1∘E1E2ςi−1∘E1E2iF1i−1ε1∘λ(2⋯r+1)∘λ(2i+1⋯i+r+1)∘λ(1⋯2i+1)\displaystyle=\sum_{r=1}^{i}\sigma^{-1}\circ E_{1}E_{2}\varsigma_{i-1}\circ E_{1}E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(2\cdots r+1)}\circ\lambda_{(2i+1\cdots i+r+1)}\circ\lambda_{(1\cdots 2i+1)} =∑r=1iE2E1ςi−1∘E2E1E2i−1F1i−1ε1∘σ−1E2i−1F1iE1∘λ(1⋯2i+1)∘λ(1⋯r)∘λ(2i⋯i+r)\displaystyle=\sum_{r=1}^{i}E_{2}E_{1}\varsigma_{i-1}\circ E_{2}E_{1}E_{2}^{i-1}F_{1}^{i-1}\varepsilon_{1}\circ\sigma^{-1}E_{2}^{i-1}F_{1}^{i}E_{1}\circ\lambda_{(1\cdots 2i+1)}\circ\lambda_{(1\cdots r)}\circ\lambda_{(2i\cdots i+r)} =∑r=1iE2E1ςi−1∘E2E1E2i−1F1i−1ε1∘λ(2⋯2i+1)∘λ(1⋯r)∘λ(2i⋯i+r)\displaystyle=\sum_{r=1}^{i}E_{2}E_{1}\varsigma_{i-1}\circ E_{2}E_{1}E_{2}^{i-1}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(2\cdots 2i+1)}\circ\lambda_{(1\cdots r)}\circ\lambda_{(2i\cdots i+r)} a44\displaystyle a_{44} =λ(12)∘E12ςi∘λ(2⋯2i+2)∘λ(1⋯2i+1)\displaystyle=\lambda_{(12)}\circ E_{1}^{2}\varsigma_{i}\circ\lambda_{(2\cdots 2i+2)}\circ\lambda_{(1\cdots 2i+1)} =E12ςi∘λ(1⋯2i+2)∘λ(1⋯2i+1)\displaystyle=E_{1}^{2}\varsigma_{i}\circ\lambda_{(1\cdots 2i+2)}\circ\lambda_{(1\cdots 2i+1)} We have b21=b12=b22=b31=b32=b33=b41=b42=b43=0b_{21}=b_{12}=b_{22}=b_{31}=b_{32}=b_{33}=b_{41}=b_{42}=b_{43}=0 b11\displaystyle b_{11} =E22ςi∘λ(2⋯2i+2)∘λ(1⋯2i+1)∘λ(2i+1,2i+2)\displaystyle=E_{2}^{2}\varsigma_{i}\circ\lambda_{(2\cdots 2i+2)}\circ\lambda_{(1\cdots 2i+1)}\circ\lambda_{(2i+1,2i+2)} =E22ςi∘λ(1⋯2i+2)∘λ(1⋯2i+1)\displaystyle=E_{2}^{2}\varsigma_{i}\circ\lambda_{(1\cdots 2i+2)}\circ\lambda_{(1\cdots 2i+1)} b13\displaystyle b_{13} =∑r=1iE22ςi−1∘E2i+1F1i−1ε1∘λ(2⋯r+1)∘λ(2i+1⋯i+r+1)∘λ(1⋯2i+1)∘E2iF1iσ−1\displaystyle=\sum_{r=1}^{i}E_{2}^{2}\varsigma_{i-1}\circ E_{2}^{i+1}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(2\cdots r+1)}\circ\lambda_{(2i+1\cdots i+r+1)}\circ\lambda_{(1\cdots 2i+1)}\circ E_{2}^{i}F_{1}^{i}\sigma^{-1} =∑r=1iE22ςi−1∘E2i+1F1i−1ε1∘λ(1⋯2i+1)∘λ(2i⋯i+r)∘λ(1⋯r)∘E2iF1iσ−1\displaystyle=\sum_{r=1}^{i}E_{2}^{2}\varsigma_{i-1}\circ E_{2}^{i+1}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(1\cdots 2i+1)}\circ\lambda_{(2i\cdots i+r)}\circ\lambda_{(1\cdots r)}\circ E_{2}^{i}F_{1}^{i}\sigma^{-1} b14\displaystyle b_{14} =∑1≤r≤i1≤s<iE22ςi−2∘E2iF1i−2ε1∘λ(2⋯s+1)(2i−1⋯i+s)∘E2iF1i−1ε1E1∘λ(1⋯r)(2i⋯i+r)∘λ(2i+1,2i+2)\displaystyle=\sum_{\begin{subarray}{c}1\leq r\leq i\\ 1\leq s<i\end{subarray}}E_{2}^{2}\varsigma_{i-2}\circ E_{2}^{i}F_{1}^{i-2}\varepsilon_{1}\circ\lambda_{(2\cdots s+1)(2i-1\cdots i+s)}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}E_{1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}\circ\lambda_{(2i+1,2i+2)} =∑1≤r≤i1≤s<iE22ςi−2∘E2iF1i−2ε1∘E2iF1i−1ε1E1∘λ(2i+1,2i+2)∘λ(2⋯s+1)(2i−1⋯i+s)∘λ(1⋯r)(2i⋯i+r)\displaystyle=\sum_{\begin{subarray}{c}1\leq r\leq i\\ 1\leq s<i\end{subarray}}E_{2}^{2}\varsigma_{i-2}\circ E_{2}^{i}F_{1}^{i-2}\varepsilon_{1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}E_{1}\circ\lambda_{(2i+1,2i+2)}\circ\lambda_{(2\cdots s+1)(2i-1\cdots i+s)}\circ\lambda_{(1\cdots r)(2i\cdots i+r)} =∑1≤r≤i1≤s<iE22ςi−2∘E2iF1i−2ε1∘E2iF1i−1ε1E1∘λ(2i−1,2i)∘λ(2⋯s+1)(2i−1⋯i+s)∘λ(1⋯r)(2i⋯i+r)\displaystyle=\sum_{\begin{subarray}{c}1\leq r\leq i\\ 1\leq s<i\end{subarray}}E_{2}^{2}\varsigma_{i-2}\circ E_{2}^{i}F_{1}^{i-2}\varepsilon_{1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}E_{1}\circ\lambda_{(2i-1,2i)}\circ\lambda_{(2\cdots s+1)(2i-1\cdots i+s)}\circ\lambda_{(1\cdots r)(2i\cdots i+r)} =∑1≤s<r≤iE22ςi−2∘E2iF1i−2ε1∘E2iF1i−1ε1E1∘λ(2⋯s+1)∘λ(1⋯r)∘λ(2i⋯i+s)∘λ(2i⋯i+r)\displaystyle=\sum_{1\leq s<r\leq i}E_{2}^{2}\varsigma_{i-2}\circ E_{2}^{i}F_{1}^{i-2}\varepsilon_{1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}E_{1}\circ\lambda_{(2\cdots s+1)}\circ\lambda_{(1\cdots r)}\circ\lambda_{(2i\cdots i+s)}\circ\lambda_{(2i\cdots i+r)} =∑1≤r′≤s′≤iE22ςi−2∘E2iF1i−2ε1∘E2iF1i−1ε1E1∘λ(2⋯r′+1)∘λ(1⋯s′+1)∘λ(2i⋯i+r′)∘λ(2i⋯i+s′+1)\displaystyle=\sum_{1\leq r^{\prime}\leq s^{\prime}\leq i}E_{2}^{2}\varsigma_{i-2}\circ E_{2}^{i}F_{1}^{i-2}\varepsilon_{1}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}E_{1}\circ\lambda_{(2\cdots r^{\prime}+1)}\circ\lambda_{(1\cdots s^{\prime}+1)}\circ\lambda_{(2i\cdots i+r^{\prime})}\circ\lambda_{(2i\cdots i+s^{\prime}+1)} b23=E2E1ςi∘λ(2⋯2i+2)∘λ(1⋯2i+1)∘E2iF1iσ−1b_{23}=E_{2}E_{1}\varsigma_{i}\circ\lambda_{(2\cdots 2i+2)}\circ\lambda_{(1\cdots 2i+1)}\circ E_{2}^{i}F_{1}^{i}\sigma^{-1} b24\displaystyle b_{24} =∑r=1iE2E1ςi−1∘λ(2⋯2i)∘E2iF1i−1ε1E1∘λ(1⋯r)(2i⋯i+r)∘λ(2i+1,2i+2)\displaystyle=\sum_{r=1}^{i}E_{2}E_{1}\varsigma_{i-1}\circ\lambda_{(2\cdots 2i)}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}E_{1}\circ\lambda_{(1\cdots r)(2i\cdots i+r)}\circ\lambda_{(2i+1,2i+2)} =∑r=1iE2E1ςi−1∘λ(2⋯2i)∘E2iF1i−1E1ε1∘λ(2i,2i+1)∘λ(1⋯r)(2i⋯i+r)\displaystyle=\sum_{r=1}^{i}E_{2}E_{1}\varsigma_{i-1}\circ\lambda_{(2\cdots 2i)}\circ E_{2}^{i}F_{1}^{i-1}E_{1}\varepsilon_{1}\circ\lambda_{(2i,2i+1)}\circ\lambda_{(1\cdots r)(2i\cdots i+r)} =∑r=1iE2E1ςi−1∘E2iF1i−1E1ε1∘λ(2⋯2i+1)∘λ(1⋯r)(2i⋯i+r)\displaystyle=\sum_{r=1}^{i}E_{2}E_{1}\varsigma_{i-1}\circ E_{2}^{i}F_{1}^{i-1}E_{1}\varepsilon_{1}\circ\lambda_{(2\cdots 2i+1)}\circ\lambda_{(1\cdots r)(2i\cdots i+r)} b34\displaystyle b_{34} =∑r=1iE1E2ςi−1∘E1E2iF1i−1ε1∘λ(2⋯r+1)∘λ(2i+1⋯i+r+1)∘λ(1⋯2i+1)∘λ(2i+1,2i+2)\displaystyle=\sum_{r=1}^{i}E_{1}E_{2}\varsigma_{i-1}\circ E_{1}E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(2\cdots r+1)}\circ\lambda_{(2i+1\cdots i+r+1)}\circ\lambda_{(1\cdots 2i+1)}\circ\lambda_{(2i+1,2i+2)} =∑r=1iE1E2ςi−1∘E1E2iF1i−1ε1∘λ(1⋯2i+2)∘λ(1⋯r)∘λ(2i⋯i+r)\displaystyle=\sum_{r=1}^{i}E_{1}E_{2}\varsigma_{i-1}\circ E_{1}E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}\circ\lambda_{(1\cdots 2i+2)}\circ\lambda_{(1\cdots r)}\circ\lambda_{(2i\cdots i+r)} =∑r=1iE1E2ςi−1∘λ(1⋯2i)∘E2iF1i−1E1ε1∘λ(2i,2i+1)∘λ(2i+1,2i+2)∘λ(1⋯r)∘λ(2i⋯i+r)\displaystyle=\sum_{r=1}^{i}E_{1}E_{2}\varsigma_{i-1}\circ\lambda_{(1\cdots 2i)}\circ E_{2}^{i}F_{1}^{i-1}E_{1}\varepsilon_{1}\circ\lambda_{(2i,2i+1)}\circ\lambda_{(2i+1,2i+2)}\circ\lambda_{(1\cdots r)}\circ\lambda_{(2i\cdots i+r)} =∑r=1iE1E2ςi−1∘λ(1⋯2i)∘E2iF1i−1ε1E1∘λ(2i+1,2i+2)∘λ(2i+1,2i+2)∘λ(1⋯r)∘λ(2i⋯i+r)\displaystyle=\sum_{r=1}^{i}E_{1}E_{2}\varsigma_{i-1}\circ\lambda_{(1\cdots 2i)}\circ E_{2}^{i}F_{1}^{i-1}\varepsilon_{1}E_{1}\circ\lambda_{(2i+1,2i+2)}\circ\lambda_{(2i+1,2i+2)}\circ\lambda_{(1\cdots r)}\circ\lambda_{(2i\cdots i+r)} =0\displaystyle=0 b44\displaystyle b_{44} =E12ςi∘λ(2⋯2i+2)∘λ(1⋯2i+1)∘λ(2i+1,2i+2)\displaystyle=E_{1}^{2}\varsigma_{i}\circ\lambda_{(2\cdots 2i+2)}\circ\lambda_{(1\cdots 2i+1)}\circ\lambda_{(2i+1,2i+2)} =E12ςi∘λ(1⋯2i+2)∘λ(1⋯2i+1)\displaystyle=E_{1}^{2}\varsigma_{i}\circ\lambda_{(1\cdots 2i+2)}\circ\lambda_{(1\cdots 2i+1)} We deduce that A=BA=B and the lemma follows. ∎