Generated by the Physics Derivation Graph. Eq. \ref{eq:1939485} is an identity. \begin{equation} \vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) \label{eq:1939485} \end{equation} Replace curl in Eq. \ref{eq:1939485} with Levi-Cevita contravariant; yields Eq. \ref{eq:9485482}. \begin{equation} \epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{ \nabla} \times \vec{E} )_k = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) \label{eq:9485482} \end{equation} Replace curl in Eq. \ref{eq:9485482} with Levi-Cevita contravariant; yields Eq. \ref{eq:2941319}. \begin{equation} \epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) \label{eq:2941319} \end{equation} Eq. \ref{eq:2934842} is an identity. \begin{equation} \epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \label{eq:2934842} \end{equation} Substitute RHS of Eq. \ref{eq:2934842} into Eq. \ref{eq:2941319}; yields Eq. \ref{eq:3949292}. \begin{equation} \left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) \label{eq:3949292} \end{equation} Simplify Eq. \ref{eq:3949292}; yields Eq. \ref{eq:3844221}. \begin{equation} \left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) \label{eq:3844221} \end{equation} Simplify Eq. \ref{eq:3844221}; yields Eq. \ref{eq:3948472}. \begin{equation} \hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) \label{eq:3948472} \end{equation} Replace summation notation in Eq. \ref{eq:3948472} with vector notation; yields Eq. \ref{eq:2109231}. \begin{equation} \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) \label{eq:2109231} \end{equation} Thus we see that LHS of Eq. \ref{eq:2109231} is equal to RHS.