Rewrite the relations (a) in the form $$\begin{array}{cc} T1 T2 Tn-2 Tn-1 Tn & \text{(a1)}\\ T_1{T}_0{T}_1{T}_0=T_0{T}_1{T}_0{T}_1\text{and}{T}_0{T}_i=T_i{T}_i,\hspace{0.17em}\text{for}\hspace{0.17em}i\in \{2,3,\dots ,n\}\text{.}& \text{(a2)}\end{array}$$

Generators (b) in terms of generators (a): $T_i=T_i,$ for $i\in \{1,2,\dots ,n\},$ and $$Y^{{\epsilon }_1^{\vee }}=T_0{T}_1\cdots T_n\cdots T_1\text{and}{Y}^{{\epsilon }_{i+1}^{\vee }}=T_i^{-1}{Y}^{{\epsilon }_i^{\vee }}{T}_i^{-1},\hspace{0.17em}\text{for}\hspace{0.17em}i\in \{1,2,\dots ,n-1\}\text{.}$$ Pictorially, $$Y^{{\epsilon }_j^{\vee }}=\begin{array}{c} j \end{array}$$

Generators (a) in terms of generators (b): $$T_0=Y^{{\epsilon }_1^{\vee }}{T}_1^{-1}\cdots T_n^{-1}\cdots T_1^{-1}\text{and}{T}_i=T_i,\hspace{0.17em}\text{for}\hspace{0.17em}i\in \{1,2,\dots ,n\}\text{.}$$

Generators (c) in terms of generators (b): If $i\in \{1,\dots ,n\}$ then $T_i=T_i$ and $$Y^{{\lambda }^{\vee }}={\left(Y^{{\epsilon }_1^{\vee }}\right)}^{{\lambda }_1}{\left(Y^{{\epsilon }_2^{\vee }}\right)}^{{\lambda }_2}\cdots {\left(Y^{{\epsilon }_n^{\vee }}\right)}^{{\lambda }_n},\text{for}\hspace{0.17em}{\lambda }^{\vee }={\lambda }_1{\epsilon }_1^{\vee }+\cdots +{\lambda }_n{\epsilon }_n^{\vee }\text{.}$$

Relations (b) from relations (a): The relations (b1) are the same as the relations (a1).

The picture $$\text{PICTURE}$$ gives $Y^{{\epsilon }_i^{\vee }}{Y}^{{\epsilon }_j^{\vee }}=Y^{{\epsilon }_j^{\vee }}{Y}^{{\epsilon }_i^{\vee }},$ for $i,j\in \{1,\dots ,n\},$ proving relations (b2).

From the picture of $Y^{{\epsilon }_j^{\vee }}$ $T_i{Y}^{{\epsilon }_j^{\vee }}=Y^{{\epsilon }_j^{\vee }}{T}_i$ for $j\in \{1,\dots ,n\}$ and $i\in \{1,\dots ,k-1\}$ with $i\ne j-1,j\text{.}$ So $$T_i{Y}^{{\epsilon }_j^{\vee }}=Y^{{\epsilon }_j^{\vee }}{T}_i,\text{for}\hspace{0.17em}j\in \{1,\dots ,n\},\hspace{0.17em}i\in \{1,\dots ,n-1\}\hspace{0.17em}\text{with}\hspace{0.17em}j\ne i,i+1,$$ proving relations (b3).

From the picture of $Y^{{\epsilon }_j^{\vee }},$ $$T_n{Y}^{{\epsilon }_j^{\vee }}=Y^{{\epsilon }_j^{\vee }}{T}_n,\text{for}\hspace{0.17em}j\in \{1,\dots ,n-1\},$$ proving relations (b4).

Relations (a) from relations (b): The relations (a1) are the same as the relations (b1).

Note that if $i\in \{2,3,\dots ,n\}$ then $$\begin{array}{ccc}{T}_{s_{\phi }}{T}_i& =& T_1\cdots T_n\cdots T_1{T}_i=T_1\cdots T_k\cdots T_{i+1}{T}_i{T}_{i-1}{T}_i{T}_{i-2}\cdots T_1\\ & =& T_1\cdots T_k\cdots T_{i+1}{T}_{i-1}{T}_i{T}_{i-1}{T}_{i-2}\cdots T_1\\ & =& T_1\cdots T_{i-2}{T}_{i-1}{T}_i{T}_{i-1}{T}_{i+1}\cdots T_k\cdots T_{i+1}{T}_i{T}_{i-1}{T}_{i-2}\cdots T_1\\ & =& T_1\cdots T_{i-2}{T}_i{T}_{i-1}{T}_i{T}_{i+1}\cdots T_k\cdots T_1=T_i{T}_1\cdots T_k\cdots T_1=T_i{T}_{s_{\phi }},\end{array}$$ which is also visible from the picture $$T_{s_{\phi }}=T_{s_{{\epsilon }_1}}=\begin{array}{c} \end{array}=T_1\cdots T_n\cdots T_1\text{.}$$

If $i\in \{2,3,\dots ,n\}$ then $T_i{T}_{s_{{\epsilon }_1}}^{-1}=T_{s_{{\epsilon }_1}}^{-1}{T}_i$ and $$T_0{T}_i=Y^{{\epsilon }_1^{\vee }}{T}_{s_{{\epsilon }_1}}^{-1}{T}_i=Y^{{\epsilon }_1^{\vee }}{T}_i{T}_{s_{{\epsilon }_1}}^{-1}=T_i{Y}^{{\epsilon }_1^{\vee }}{T}_{s_{{\epsilon }_1}}^{-1}=T_i{T}_0$$ which proves the second set of relations in (a2).

Since $Y^{{\epsilon }_1^{\vee }}=T_0{T}_1\cdots T_n\cdots T_1$ and $Y^{{\epsilon }_2^{\vee }}=T_1^{-1}{Y}^{{\epsilon }_1^{\vee }}{T}_1^{-1}=T_1^{-1}{T}_0{T}_1\cdots T_n\cdots T_2$ we have $$Y^{{\epsilon }_1^{\vee }}{Y}^{{\epsilon }_2^{\vee }}=T_0{T}_1\cdots T_n\cdots T_2{T}_0{T}_1\cdots T_n\cdots T_2=T_0{T}_1{T}_0{T}_2\cdots T_n\cdots T_2{T}_1\cdots T_n\cdots T_2\text{.}$$ Then $T_1^{-1}{Y}^{{\epsilon }_1^{\vee }}{Y}^{{\epsilon }_2^{\vee }}=T_1^{-1}{Y}^{{\epsilon }_1^{\vee }}{Y}^{{\epsilon }_2^{\vee }}=T_1^{-1}{Y}^{{\epsilon }_1^{\vee }}{T}_1^{-1}{Y}^{{\epsilon }_1^{\vee }}{T}_1^{-1}=Y^{{\epsilon }_2^{\vee }}{Y}^{{\epsilon }_1^{\vee }}{T}_1^{-1}=Y^{{\epsilon }_1^{\vee }}{Y}^{{\epsilon }_2^{\vee }}{T}_1^{-1}$ gives $$\begin{array}{ccc}{T}_1{Y}^{{\epsilon }_1^{\vee }}{Y}^{{\epsilon }_2^{\vee }}& =& T_1{T}_0{T}_1{T}_0{T}_2\cdots T_n\cdots T_2{T}_1\cdots T_n\cdots T_2\\ =Y^{{\epsilon }_1^{\vee }}{Y}^{{\epsilon }_2^{\vee }}{T}_1& =& T_0{T}_1{T}_0{T}_2\cdots T_n\cdots T_2{T}_{s_{\phi }}=T_0{T}_1{T}_0{T}_{s_{\phi }}{T}_2\cdots T_n\cdots T_2\\ & =& T_0{T}_1{T}_0{T}_1{T}_2\cdots T_n\cdots T_2{T}_1\cdots T_n\cdots T_2,\end{array}$$ since $T_{s_{\phi }}{T}_i=T_i{T}_{s_{\phi }}$ for $i=2,\dots ,n\text{.}$ Multiplying on the right by ${\left(T_2\cdots T_n\cdots T_2{T}_1\cdots T_n\cdots T_2\right)}^{-1}$ gives $T_1{T}_0{T}_1{T}_0=T_0{T}_1{T}_0{T}_1\text{.}$ This establishes the first relation in (a2).

Relations (c) from relations (b): Relations (c1) are the same as relations (b1). The first set of relations in (c2) are equivalent to the relations in (b2).

If $i\in \{1,\dots ,n-1\}$ and ${\lambda }^{\vee }={\lambda }_1{\epsilon }_1^{\vee }+\cdots +{\lambda }_n{\epsilon }_n^{\vee }$ then $$⟨{\lambda }^{\vee },{\alpha }_i⟩=⟨{\lambda }^{\vee },e_i-{\epsilon }_{i+1}⟩={\lambda }_i-{\lambda }_{i+1}=\{\begin{array}{cc}0,& \text{if}\hspace{0.17em}{\lambda }_i={\lambda }_{i+1},\\ 1,& \text{if}\hspace{0.17em}{\lambda }_{i+1}={\lambda }_i-1,\end{array}$$ and $$⟨{\lambda }^{\vee },{\alpha }_n⟩==⟨{\lambda }^{\vee },2{\epsilon }_n⟩=2{\lambda }_n=\{\begin{array}{cc}0,& \text{if}\hspace{0.17em}{\lambda }_n=0,\\ 1,& \text{never.}\end{array}$$ Thus the second set of relations in (c2) are equivalent to $$\begin{array}{cc}{T}_i\left(Y^{{\epsilon }_i^{\vee }}{Y}^{{\epsilon }_{i+1}^{\vee }}\right)={\left(Y^{{\epsilon }_i^{\vee }}{Y}^{{\epsilon }_{i+1}^{\vee }}\right)}^{{\lambda }_i}{T}_i,& \text{(c3)}\\ T_i{Y}^{{\epsilon }_j^{\vee }}=Y^{{\epsilon }_j^{\vee }}{T}_i,\text{for}\hspace{0.17em}j\in \{1,\dots ,n\}\hspace{0.17em}\text{with}\hspace{0.17em}j\ne i,i+1,& \text{(c4)}\\ T_i^{-1}{\left(Y^{{\epsilon }_i^{\vee }}\right)}^{{\lambda }_i}{\left(Y^{{\epsilon }_{i+1}^{\vee }}\right)}^{{\lambda }_i-1}={\left(Y^{{\epsilon }_i^{\vee }}\right)}^{{\lambda }_i-1}{\left(Y^{{\epsilon }_{i+1}^{\vee }}\right)}^{{\lambda }_i}{T}_i,& \text{(c5)}\end{array}$$ for $i\in \{1,\dots ,n-1\},$ and $$\begin{array}{cc}{T}_n{Y}^{{\epsilon }_j}=Y^{{\epsilon }_j^{\vee }}{T}_n,\text{if}\hspace{0.17em}j\in \{1,\dots ,n-1\}\text{.}& \text{(c6)}\end{array}$$ Relations (c4) and (c6) are relations (b3) and (b4) and the computations $$T_i^{-1}{Y}^{{\epsilon }_i^{\vee }}{Y}^{{\epsilon }_{i+1}^{\vee }}=T_i^{-1}{Y}^{{\epsilon }_i^{\vee }}{T}_i^{-1}{Y}^{{\epsilon }_i^{\vee }}{T}_i^{-1}=Y^{{\epsilon }_{i+1}^{\vee }}{Y}^{{\epsilon }_i^{\vee }}{T}_i^{-1},$$ and $$\begin{array}{ccc}{T}_i^{-1}{\left(Y^{{\epsilon }_i^{\vee }}\right)}^{{\lambda }_i}{\left(Y^{{\epsilon }_{i+1}^{\vee }}\right)}^{{\lambda }_i-1}& =& T_i^{-1}{Y}^{{\epsilon }_i^{\vee }}{\left(Y^{{\epsilon }_i^{\vee }}\right)}^{{\lambda }_i-1}{\left(Y^{{\epsilon }_{i+1}^{\vee }}\right)}^{{\lambda }_i-1}\\ & =& T_i^{-1}{Y}^{{\epsilon }_i^{\vee }}{T}_i^{-1}{T}_i{\left(Y^{{\epsilon }_i^{\vee }}\right)}^{{\lambda }_i-1}{\left(Y^{{\epsilon }_{i+1}^{\vee }}\right)}^{{\lambda }_i-1}\\ & =& Y^{{\epsilon }_{i+1}^{\vee }}{T}_i{\left(Y^{{\epsilon }_i^{\vee }}\right)}^{{\lambda }_i-1}{\left(Y^{{\epsilon }_{i+1}^{\vee }}\right)}^{{\lambda }_i-1}\\ & =& Y^{{\epsilon }_{i+1}^{\vee }}{\left(Y^{{\epsilon }_i^{\vee }}\right)}^{{\lambda }_i-1}{\left(Y^{{\epsilon }_{i+1}^{\vee }}\right)}^{{\lambda }_i-1}{T}_i\\ & =& {\left(Y^{{\epsilon }_i^{\vee }}\right)}^{{\lambda }_i-1}{\left(Y^{{\epsilon }_{i+1}^{\vee }}\right)}^{{\lambda }_i}{T}_i,\end{array}$$ establish the relations in (c3) and (c5).

Relations (b) from relations (c): The relations (b1) are the same as the relations (c1).

The relations in (b2) are equivalent to the first set of relations in (c2).

In view of the equivalence of the second set of relations in (c2) with the relations in (c3-6), the relations in (b2), (b3) and (b4) are a subset of the relations in (c2).

$\square$