I am trying to prove the following, from Benedetti and Risler's book:
The "above proposition" is:
It seems an easy proposition that boils down to taking the complement of the complement of F, since the complement of $\{x \in \mathbb{R}^{n}: P(x)>0\}$ is $\{x \in \mathbb{R}^{n}: P(x) \leq 0\}$. But I end up with a finite intersection of unions of sets of this form, and I can't rewrite it as the intended form. Any help is appreciated.