Generated by the Physics Derivation Graph. Eq. \ref{eq:3849595} is an initial equation. \begin{equation} x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle \label{eq:3849595} \end{equation} Apply operator in Eq. \ref{eq:3849595} to ket; yields Eq. \ref{eq:4940359}. \begin{equation} x = \langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle \label{eq:4940359} \end{equation} Apply operator in Eq. \ref{eq:3849595} to bra; yields Eq. \ref{eq:4349300}. \begin{equation} x = \langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle \label{eq:4349300} \end{equation} Simplify Eq. \ref{eq:4940359}; yields Eq. \ref{eq:2409402}. \begin{equation} x = a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle \label{eq:2409402} \end{equation} Simplify Eq. \ref{eq:4349300}; yields Eq. \ref{eq:4934893}. \begin{equation} x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle \label{eq:4934893} \end{equation} LHS of Eq. \ref{eq:2409402} is equal to LHS of Eq. \ref{eq:4934893}; yields Eq. \ref{eq:3495045}. \begin{equation} a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle \label{eq:3495045} \end{equation} Subtract \(a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle\) from both sides of Eq. \ref{eq:3495045}; yields Eq. \ref{eq:4939583}. \begin{equation} a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0 \label{eq:4939583} \end{equation} Combine like terms in Eq. \ref{eq:4939583}; yields Eq. \ref{eq:3494855}. \begin{equation} ( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0 \label{eq:3494855} \end{equation} Eq. \ref{eq:3494855} is one of the final equations.