$H$ Hilberspace, $K_1$, $K_2$ convex, closed subset of H. $K_1\subset K_2$
I would like to show that for all $x\in H$: $ \|P_{K_1}(x)-P_{K_2}(x)\|^2 \leq 2(d(x,K_1)^2-d(x,K_2)^2) $
My first idea was to add and substract x on the left side $ \|P_{K_1}(x)-P_{K_2}(x)-x+x \|^2 \leq \| P_{K_1}(x)-x\|^2+\|x-P_{K_2}(x)\|^2 $
Can I do that? How is it possible to continue?