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How to show that the sinc function series $\sum_{n=-\infty}^\infty\text{sinc}(x)$ is equal to pi for all $x$?

We know the very well-known identity: $$\sum_{n=-\infty}^\infty\text{sinc}(n)=\pi.$$ But how to show that $$\sum_{n=-\infty}^\infty\text{sinc}(x+n)=\pi?$$ In other words, how to prove that $$\sum_{n=-\infty}^\infty\text{sinc}(x)=\pi, \qquad \forall x?$$ Any ideas?