Does anybody know how to derive
$$\det(\mathbf A)\cdot \det(\mathbf D + \mathbf E \cdot \mathbf A^{-1} \cdot \mathbf B) = \det(\mathbf D)\cdot \det(\mathbf A + \mathbf B \cdot \mathbf D^{-1} \cdot \mathbf E)$$
where $\mathbf A$, $\mathbf B$, $\mathbf C$, $\mathbf D$, $\mathbf E$ are non-singular matrices?
Most likely, it requires the Sylvester determinant theorem).