$(a,t_{\alpha_{i}}+1)$, we can replace all occurrences of the timed letters of
the form $(a,t_{\alpha_{i}}+1)$ in $w$ with $(a,t_{\alpha_{i}}+\varepsilon)$
for some $\varepsilon>0$ such that

  1. 1.

$t_{\alpha_{i}}+\varepsilon\neq t_{\alpha_{j}}$ for any
$j\in[n]\setminus\\{i\\}$,

  2. 2.

$w^{\prime}$ is still accepted in ${\mathcal{T}}^{\prime}$ by an accepting run
$\rho^{\prime\prime}$ that visits the same states $\rho^{\prime}$.

The number of good pairs in $w^{\prime}$ now is $n-1$. We can iteratively
repeat the above procedure on $w^{\prime}$ to get a word $\hat{w}$ that has
exactly $p$ good pairs and is accepted by ${\mathcal{T}}^{\prime}$. As $p<k$,
however, $\hat{w}$ is not in $L_{k}$, and hence, $L({\mathcal{T}}^{\prime})$
is not a subset of $L({\mathcal{T}})$.


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