9. Let $f: \mathbf{R} \rightarrow \mathbf{R}$ satisfy: for any real numbers $x, y$, we have $f(2 x)+f(2 y)=f(x+y) f(x-y)$, and $f(\pi)=0$, and $f(x)$ is not identically zero, then the period of $f(x)$ is $\qquad$.
9. $4 \pi$ Hint: First, it is easy to prove that $f(x)$ is an even function. Then let $x=t+\frac{\pi}{2}, y=t-\frac{\pi}{2}$, we have $f(2 t+\pi)=-f(2 t-\pi)$, thus $f(x+4 \pi)=-f(x+2 \pi)=f(x)$.