Given a direct mechanism, we define a buyer's expected utility $u(\theta)$ conditional on her type being $\theta$ by $u(\theta)=\theta q(\theta)-t(\theta)$, where $q:[\underline{\theta},\bar{\theta}]\to[0,1]$ and $t:[\underline{\theta},\bar{\theta}]\to\mathbb{R}$.
We also define that a direct mechanism is incentive-compatible if truth-telling is optimal for every $\theta\in[\underline{\theta},\bar{\theta}]$, i.e., $$u(\theta)\geq \theta q(\theta')-t(\theta'),\quad\forall\theta,\theta'\in[\underline{\theta},\bar{\theta}].$$
LEMMA: For an incentive-compatible direct mechanism, we want to show that for all $\theta$ that $u$ is differentiable, we have $u'(\theta)=q(\theta)$.
PROOF: Consider any $\theta$ for which $u$ is differentiable. Let $\delta>0$. Then by incentive compatibility, we have the following:
I don't understand how from incentive compatibility it follows the first and second inequality 2.6 and 2.8. How can we use the same $\theta$ in the proof while the definition clearly states $u(\theta)\geq \theta q(\theta')-t(\theta'),\forall\theta,\theta'$?