This leads to $$ \sum_{P'H'}\langle \Phi_H^P | \hat{H} |\Phi_{H'}^{P'} \rangle C_{H'}^{P'}-\lambda C_H^{P}=0, $$ for all sets of \( P \) and \( H \).

If we then multiply by the corresponding \( C_H^{*P} \) and sum over \( PH \) we obtain $$ \sum_{PP'HH'}C_H^{*P}\langle \Phi_H^P | \hat{H} |\Phi_{H'}^{P'} \rangle C_{H'}^{P'}-\lambda\sum_{PH}|C_H^P|^2=0, $$ leading to the identification \( \lambda = E \). This means that we have for all \( PH \) sets $$ \begin{equation} \sum_{P'H'}\langle \Phi_H^P | \hat{H} -E|\Phi_{H'}^{P'} \rangle = 0. \tag{1} \end{equation} $$