Nested Radon-Nikodym Derivative

Question

Let $\pi$ and $\eta$ be two measures on the same measurable space $(E, \mathcal{E})$ and that $\pi\ll\eta$ so the Radon-Nikodym derivative exists $$ \frac{d \pi}{d \eta} = \rho $$ Assuming all regularity conditions necessary (e.g. $\pi \ll \lambda$ and $\eta \ll \lambda$) can I say $$ \frac{d \pi}{d \eta} = \frac{\displaystyle \frac{d\pi}{d \lambda}}{\displaystyle \frac{d \eta}{d\lambda}} = \frac{\rho_\pi}{\rho_\eta} $$ where $\rho_\pi$ and $\rho_\eta$ are the densities of $\pi$ and $\eta$ respectively with respect to $\lambda$?