The evenness or oddness of the number of transpositions is determined only by the permutation. (In "Linear Algebra" by Ichiro Satake.)

Question

I am reading "Linear Algebra" by Ichiro Satake.

The following theorem is in this book:

Any permutation can be expressed as the product of transpositions. In doing so, the evenness or oddness of the number of transpositions is determined only by the permutation, independently of the method of expression.

The following is his proof.

Satake's proof is here.

He proved if there exists at least one nonzero polynomial such that $\sigma f=\pm f$ for any $\sigma$ and $\sigma_1 f = -f$ for at least one $\sigma_1$, then the above theorem holds.
He gave us the difference product as an example of a nonzero alternating polynomial.

When I read this proof for the first time, I thought it was an interesting proof (an interesting approach).

But if we know the difference product is a nonzero alternating polynomial, it is very easy to prove the above theorem by the properties of the difference product.　　

So, I think the first half of his argument is unnecessary to prove the above theorem.

Why did Satake write such a long proof?