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$\begingroup$I guess this is off-topic here, but it's a one-line proof if you substitute for each $f,g,h...$ the Fourier integral $f(x)=\int \hat{f}(\xi)e^{i\xi x}d\xi$, and then integrate over $x$, using that $\int_{-\infty}^\infty e^{ix(\xi_1+\xi_2\cdots)}dx=2\pi\delta(\xi_1+\xi_2\cdots)$.$\endgroup$

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