Woodbury matrix inversion prove

Question

I'm trying to solve Woodbury matrix inversion to prove the correctness of this result.

\begin{align} &(A+UCV) \left(A^{-1}-A^{-1}U(C^{-1}+VA^{-1}U)^{-1}VA^{-1}\right)=\\ &=AA^{-1}-AA^{-1}U(C^{-1}+VA^{-1}U)^{-1}VA^{-1}+UCVA^{-1}-UCVA^{-1}U(C^{-1}+VA^{-1}U)^{-1}VA^{-1} \\ &=I-U(C^{-1}+VA^{-1}U)^{-1}VA^{-1}+UCVA^{-1}+U(C^{-1}+VA^{-1}U)^{-1}VA^{-1}-UCVA^{-1} \\ &=I \end{align} I don't understand the next step: \begin{align} -UCVA^{-1}U(C^{-1}+VA^{-1}U)^{-1}VA^{-1}&=-UC(-C^{-1}+C^{-1}+VA^{-1}U)(C^{-1}+VA^{-1}U)^{-1}VA^{-1} \\ &=U(C^{-1}+VA^{-1}U)^{-1}VA^{-1}-UCVA^{-1}. \end{align}

I've tried to understand this: Comment from a partner

But I only understood this: From the first summand, the two minus signs cancel and \begin{align} UCC^{-1}=U. \end{align}, so you get \begin{align} U(C^{-1}+VA^{-1}U)^{-1}VA^{-1}. \end{align}

(English is not my native language, apology and thanks)

Answer 1

$-UC(-C^{-1}+C^{-1}+VA^{-1}U)(C^{-1}+VA^{-1}U)^{-1}VA^{-1} = \\ =(-UC)(-C^{-1})(C^{-1}+VA^{-1}U)^{-1}VA^{-1} + (-UC)(C^{-1}+VA^{-1}U)(C^{-1}+VA^{-1}U)^{-1}VA^{-1}=\\ =U(C^{-1}+VA^{-1}U)^{-1}VA^{-1}-UCVA^{-1}$