No, they just used sloppy notations. They omitted what we called "attainment argument".
Suppose the attainment of supremum occurs at $h_{0}\in\mathcal{H}$,$$Pr\left[sup_{h\in\mathcal{H}}\left|\widehat{E}(h,S_{1})-\widehat{E}(h,S_{2})\right|>\frac{\epsilon}{2}\right]=Pr\left[\left|\widehat{E}(h_{0},S_{1})-\widehat{E}(h_{0},S_{2})\right|>\frac{\epsilon}{2}\right]\leq_{1-2\delta}\text{(by Lemma 3.5)}Pr\left[\left|\widehat{E}(h_0',S'_{1})-\widehat{E}(h_0',S'_{2})\right|>\rho(h_{0})\right]\leq Pr\left[sup_{h'\in\mathcal{H}}\left|\widehat{E}(h',S'_{1})-\widehat{E}(h',S'_{2})\right|>\rho(h_{0})\right]$$ is what they argued.