Let $A, B \in M_n, A^2=A, B^2=B$
Prove that $\operatorname{rank}(A-B)=\operatorname{rank}(A-AB)+\operatorname{rank}(B-AB)$
My attempt: I use Sysvester's rank inequality
$\operatorname{rank}(A-B)=\operatorname{rank}(A-AB+AB-B)\le \operatorname{rank}(A-AB)+\operatorname{rank}(AB-B)=\operatorname{rank}(A-AB)+\operatorname{rank}(B-AB)$
But I don't know how to prove $\operatorname{rank}(A-B)\ge\operatorname{rank}(A-AB)+\operatorname{rank}(B-AB)$