Proof of Minsky Papert Symmetrization technique

Question

I frequently hear about the Minsky-Papert Symmetrization technique in many papers with a reference to the book of Minsky. I could not locate the book online. Could someone supply me a proof of the symmetrization technique?

For instance, it is used in Lemma $5$ in this paper http://www.csee.usf.edu/~tripathi/Publication/polynomial-degree-conference.pdf