Generated by the Physics Derivation Graph. Eq. \ref{eq:3402919} is an initial equation. \begin{equation} \langle \psi| \hat{A} |\psi \rangle = \langle a \rangle \label{eq:3402919} \end{equation} Conjugate transpose of both sides of Eq. \ref{eq:3402919}; yields Eq. \ref{eq:2495954}. \begin{equation} \left(\langle\psi| \hat{A} |\psi \rangle \right)^+ = \left(\langle a \rangle\right)^+ \label{eq:2495954} \end{equation} Distribute conjugate transpose to factors in Eq. \ref{eq:2495954}; yields Eq. \ref{eq:2390094}. \begin{equation} \langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^* \label{eq:2390094} \end{equation} Eq. \ref{eq:2484892} is an assumption. \begin{equation} \hat{A}^+ = \hat{A} \label{eq:2484892} \end{equation} Substitute RHS of Eq. \ref{eq:2484892} into Eq. \ref{eq:2390094}; yields Eq. \ref{eq:2494040}. \begin{equation} \langle \psi| \hat{A} |\psi \rangle = \langle a \rangle^* \label{eq:2494040} \end{equation} Substitute RHS of Eq. \ref{eq:2494040} into Eq. \ref{eq:3402919}; yields Eq. \ref{eq:4930585}. \begin{equation} \langle a \rangle^* = \langle a \rangle \label{eq:4930585} \end{equation} Eq. \ref{eq:4930585} is one of the final equations.