Apply the structural dichotomy to the complement of a $P_5$-free graph. Strong induction reduces the proof to a critical graph. A large restricted set and a uniform blockade each contradict criticality, the latter by summing the blockwise values of $\kappa$. Finally, $\kappa(G)\leq \max\{\alpha(G),\omega(G)\}^2$ gives the stated Erdős–Hajnal bound.