How can I prove $A − A(A + B)^{−1}A = B − B(A + B)^{−1}B$ for matrices $A$ and $B$? The matrix cookbook (page 16) offers this amazing result:
$$A − A(A + B)^{−1}A = B − B(A + B)^{−1}B$$
This seems to be too unbelievable to be true and I can't seem to prove it. Can anyone verify this equation/offer proof?

\begin{align}
A - A(A+B)^{-1}A & = A(A+B)^{-1}(A+B) - A(A+B)^{-1}A \\ &= A(A+B)^{-1}(A+B - A)\\ &= A(A+B)^{-1}B \\ &= (A+B - B)(A+B)^{-1}B \\ &= (A+B)(A+B)^{-1}B - B(A+B)^{-1}B \\ &= B - B(A+B)^{-1}B
\end{align}
