\begin{aligned} \Phi\left(Z^{\prime}\right)-\Phi(Z) &=\sup _{f \in \mathcal{F}} \left(\mathbb{E}[f]-\widehat{E}_{Z^{\prime}}(f)\right)-\sup _{f \in \mathcal{F}} \left(\mathbb{E}[f]-\widehat{E}_{Z}(f)\right) \\ & \leqslant \sup _{f \in \mathcal{F}} \left(\widehat{E}_{Z}(f)-\widehat{E}_{Z^{\prime}}(f)\right) \\ &=\sup_{f\in\mathcal{F}}\frac{\sum^m_{i=1}f(z_i)-\sum^m_{i=1}f(z^\prime_i)}{m}\\&=\sup _{f \in \mathcal{F}} \frac{f\left(z_{m}\right)-f\left(z_{m}^{\prime}\right)}{m} \\ & \leqslant \frac{1}{m} \end{aligned}
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