How can I prove $A − A(A + B)^{−1}A = B − B(A + B)^{−1}B$ for matrices $A$ and $B$?

Question

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?

user1551 · Answer

You may simply put $X=A+B$ and show that
begin{aligned}
A-AX^{-1}A
&=(X-B)-(X-B)X^{-1}(X-B)\
&=(X-B)-(X-2B+BX^{-1}B)\
&=B-BX^{-1}B.
end{aligned}

Kenny Wong · Answer

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}

How can I prove $A − A(A + B)^{−1}A = B − B(A + B)^{−1}B$ for matrices $A$ and $B$?

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