Given $\mathrm{A} \in \mathbb{F}^{n \times n}, \; \mathrm{B} \in \mathbb{F}^{n \times k}, \; \mathrm{C} \in \mathbb{F}^{k \times k}, \; \mathrm{D} \in \mathbb{F}^{k \times n}$, and $\mathrm{A, \;C}$ are invertible.
Prove that $\mathrm{\left(A + BCD\right)^{-1} = A^{-1} - A^{-1}B\left(C^{-1}+DA^{-1}B\right)^{-1}DA^{-1}}$
I proved it by showing,
$$ \mathrm{\left(A + BCD\right)^{-1}\left( A^{-1} - A^{-1}B\left(C^{-1}+DA^{-1}B\right)^{-1}DA^{-1} \right) - I = O} $$, and the solution in the textbook support this idea.
But why isn't it necessary to prove $$ \mathrm{\left( A^{-1} - A^{-1}B\left(C^{-1}+DA^{-1}B\right)^{-1}DA^{-1} \right)\left(A + BCD\right)^{-1} - I = O} $$ also?
