I have to prove this equivalence $$(zI-E)^{-1}MC(zI-\Phi)^{-1}=T(zI-\Phi)^{-1}-(zI-E)^{-1}T$$
where the matrices $E,\Phi$ (squared) and $T$ satisfy
$$T\Phi-ET-MC=0$$
the matrix $I$ is the identity, while $z$ is a scalar.
I have tried by substitute $MC$ with $T\Phi-ET$, but this don't bring immediately to the solution. In fact I get
$$(zI-E)^{-1}MC(zI-\Phi)^{-1}=(zI-E)^{-1}T\Phi(zI-\Phi)^{-1}-(zI-E)^{-1}ET(zI-\Phi)^{-1}$$
how can I carry on the calculus?