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