Decomposition of Boolean functions into the sum of products of subfunctions

We obtain a theorem on the representation of Boolean functions in the polynomial form
$$ f(x,y)=\sum_\sigma\sum_\tau\alpha_{\tau\sigma}f(\tau,y)f(x,\sigma), $$
where the Boolean summations are taken over all Boolean vectors $\sigma$ and $\tau$, $\alpha_{\tau\sigma}\in\{0,1\}$, $x$ and $y$ are collections of Boolean variables. We also give a method for finding the coefficients $\alpha_{\tau\sigma}$.