[proofplan]
We prove the two cases directly from the definition of the standard Hecke basis and the quadratic relation. In the length-increasing case, appending $s_i$ to a reduced expression for $w$ gives a reduced expression for $ws_i$, so the product is the basis element $T_{ws_i}$. In the length-decreasing case, we write $v=ws_i$, so that $w=vs_i$ and the first case gives $T_w=T_vT_i$; multiplying once more by $T_i$ reduces the claim to the defining quadratic relation.
[/proofplan]
[step:Use the standard basis convention for reduced words]
The ring $R=\mathbb Z[q]$ is a commutative ring with identity, and the element $q\in R$ is the Hecke parameter. By the standard basis construction for type $A$ Hecke algebras, as in [citetheorem:8464], if
\begin{align*}
w=s_{j_1}s_{j_2}\cdots s_{j_m}
\end{align*}
is a reduced expression in $S_n$, where $m=\ell(w)$ and each $j_a\in\{1,\dots,n-1\}$, then
\begin{align*}
T_w=T_{j_1}T_{j_2}\cdots T_{j_m}.
\end{align*}
This element is independent of the chosen reduced expression. For the identity permutation $1_{S_n}$, the reduced expression is empty and the corresponding product is the multiplicative identity of $\mathcal H_q(S_n)$.
[/step]
[step:Prove the formula when right multiplication increases length]
Assume that $w\in S_n$ and $i\in\{1,\dots,n-1\}$ satisfy
\begin{align*}
\ell(ws_i)=\ell(w)+1.
\end{align*}
Choose a reduced expression
\begin{align*}
w=s_{j_1}s_{j_2}\cdots s_{j_m},
\end{align*}
where $m=\ell(w)$. Then
\begin{align*}
ws_i=s_{j_1}s_{j_2}\cdots s_{j_m}s_i.
\end{align*}
This expression has length $m+1$, and by the hypothesis
\begin{align*}
m+1=\ell(w)+1=\ell(ws_i).
\end{align*}
Therefore it is a reduced expression for $ws_i$. Using the definition of $T_w$ from reduced expressions, we obtain
\begin{align*}
T_{ws_i}=T_{j_1}T_{j_2}\cdots T_{j_m}T_i.
\end{align*}
Since $T_w=T_{j_1}T_{j_2}\cdots T_{j_m}$, this gives
\begin{align*}
T_wT_i=T_{ws_i}.
\end{align*}
[guided]
The point of the length-increasing hypothesis is that it certifies that appending $s_i$ does not introduce a cancellation in the Coxeter word. Choose a reduced expression
\begin{align*}
w=s_{j_1}s_{j_2}\cdots s_{j_m},
\end{align*}
where $m=\ell(w)$ and each $j_a$ lies in $\{1,\dots,n-1\}$. Multiplying on the right by $s_i$ gives the expression
\begin{align*}
ws_i=s_{j_1}s_{j_2}\cdots s_{j_m}s_i.
\end{align*}
This expression has exactly $m+1$ simple transpositions. The hypothesis says
\begin{align*}
\ell(ws_i)=\ell(w)+1=m+1.
\end{align*}
Since the Coxeter length is the minimum possible number of simple transpositions in an expression for the permutation, an expression for $ws_i$ with exactly $\ell(ws_i)$ factors is reduced.
Now we translate this reduced word statement into the Hecke algebra. The standard basis convention gives
\begin{align*}
T_w=T_{j_1}T_{j_2}\cdots T_{j_m}
\end{align*}
and, because the appended expression is reduced,
\begin{align*}
T_{ws_i}=T_{j_1}T_{j_2}\cdots T_{j_m}T_i.
\end{align*}
Substituting the first equality into the second yields
\begin{align*}
T_{ws_i}=T_wT_i.
\end{align*}
Thus, in the length-increasing case,
\begin{align*}
T_wT_i=T_{ws_i}.
\end{align*}
[/guided]
[/step]
[step:Reduce the length-decreasing case to the quadratic relation]
Assume that $w\in S_n$ and $i\in\{1,\dots,n-1\}$ satisfy
\begin{align*}
\ell(ws_i)=\ell(w)-1.
\end{align*}
Define
\begin{align*}
v:=ws_i\in S_n.
\end{align*}
Since $s_i^2=1_{S_n}$, we have
\begin{align*}
w=vs_i.
\end{align*}
Also,
\begin{align*}
\ell(v)=\ell(ws_i)=\ell(w)-1,
\end{align*}
so
\begin{align*}
\ell(vs_i)=\ell(w)=\ell(v)+1.
\end{align*}
Applying the length-increasing case to $v$ and $i$ gives
\begin{align*}
T_vT_i=T_{vs_i}=T_w.
\end{align*}
Therefore
\begin{align*}
T_wT_i=(T_vT_i)T_i=T_vT_i^2.
\end{align*}
By the quadratic relation in $\mathcal H_q(S_n)$,
\begin{align*}
T_i^2=(q-1)T_i+q.
\end{align*}
Multiplying this identity on the left by $T_v$ gives
\begin{align*}
T_vT_i^2=(q-1)T_vT_i+qT_v.
\end{align*}
Using $T_vT_i=T_w$ and $v=ws_i$, we obtain
\begin{align*}
T_wT_i=(q-1)T_w+qT_{ws_i}.
\end{align*}
Since $R=\mathbb Z[q]$ is commutative, the two summands may be reordered, so
\begin{align*}
T_wT_i=qT_{ws_i}+(q-1)T_w.
\end{align*}
This is the desired formula in the length-decreasing case.
[/step]
