Generated by the Physics Derivation Graph. Eq. \ref{eq:4978059} is an initial equation. \begin{equation} \vec{p}_{\rm before} = \vec{p}_{\rm after} \label{eq:4978059} \end{equation} Eq. \ref{eq:2840008} is an initial equation. \begin{equation} \vec{p}_{\rm before} = \vec{p}_{1} \label{eq:2840008} \end{equation} Eq. \ref{eq:1209604} is an initial equation. \begin{equation} \vec{p}_{\rm after} = \vec{p}_{2}+\vec{p}_{electron} \label{eq:1209604} \end{equation} Substitute LHS of Eq. \ref{eq:2840008} into Eq. \ref{eq:4978059}; yields Eq. \ref{eq:2491904}. \begin{equation} \vec{p}_{\rm after} = \vec{p}_{1} \label{eq:2491904} \end{equation} Substitute LHS of Eq. \ref{eq:2491904} into Eq. \ref{eq:1209604}; yields Eq. \ref{eq:5610925}. \begin{equation} \vec{p}_{1} = \vec{p}_{2}+\vec{p}_{electron} \label{eq:5610925} \end{equation} Subtract \(\vec{p}_{2}\) from both sides of Eq. \ref{eq:5610925}; yields Eq. \ref{eq:4068150}. \begin{equation} \vec{p}_{1}-\vec{p}_{2} = \vec{p}_{electron} \label{eq:4068150} \end{equation} Swap LHS of Eq. \ref{eq:4068150} with RHS; yields Eq. \ref{eq:4200334}. \begin{equation} \vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2} \label{eq:4200334} \end{equation} Multiply Eq. \ref{eq:4200334} by Eq. \ref{eq:4200334}; yields Eq. \ref{eq:4218805}. \begin{equation} \vec{p}_{electron}\cdot\vec{p}_{electron} = ( \vec{p}_{1}\cdot\vec{p}_{1})+( \vec{p}_{2}\cdot\vec{p}_{2})-2( \vec{p}_{1}\cdot\vec{p}_{2}) \label{eq:4218805} \end{equation}