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\end{equation}  The first part of the theorem is proven.  For proving the second part of the theorem, we have to show that whenever uniform one-sided convergence of mean to their mathematical expectations takes place, the ERM principle is strictly consistent, that is, taht that  for any $\epsilon$ the convergence \begin{equation}  P\bigg \{ \bigg | \inf_{a \in \Lambda(c)} \int Q(z,a)dF(z) - \inf_{a \in \Lambda(c)} \frac{1}{l} \sum_{i=1}^{l}Q(z_i,a)\bigg | > \epsilon \bigg \} \xrightarrow[l \rightarrow \infty]{} 0  \end{equation}