TeX source:

\begin{split} \mathbf{F}_1\rho\mathbf{F}_1^\dagger + \mathbf{F}_2\rho\mathbf{F}_2^\dagger &= \frac{1}{2}(\mathbf{E}_1\rho\mathbf{E}_1^\dagger + \mathbf{E}_1\rho\mathbf{E}_2^\dagger + \mathbf{E}_2\rho\mathbf{E}_1^\dagger + \mathbf{E}_2\rho\mathbf{E}_2^\dagger) \\&+ \frac{1}{2}(\mathbf{E}_1\rho\mathbf{E}_1^\dagger - \mathbf{E}_1\rho\mathbf{E}_2^\dagger - \mathbf{E}_2\rho\mathbf{E}_1^\dagger + \mathbf{E}_2\rho\mathbf{E}_2^\dagger) = \mathbf{E}_1\rho\mathbf{E}_1^\dagger + \mathbf{E}_2\rho\mathbf{E}_2^\dagger. \end{split}