Let $f=[f_0,\dots,f_n]$ (not necessarily defined at $P$), and let $g=[g_0,\dots,g_n]$ and $h=[h_0,\dots,h_n]$ be defined at $P$, s.t. for each $i,j$, $f_ig_j=f_jg_i$ and $f_ih_j=f_jh_i$. Since $g$ and $h$ are regular at $P$, there exist $i,j$ s.t. $g_i(P)\neq 0$ and $h_j(P)\neq 0$. We want to verify that $g(P)=h(P)$. Note that (by definition; this isn't written explicitly, but it should be), $f\neq 0$, so for some $k$ we have $f_k\neq 0$. Now note that since $f_k g_i=f_i g_k$, we have that $f_i\neq 0$, and therefore since we also have $f_i h_j= f_j h_i$, we have that $h_i\neq 0$. Since we a…

So either 9b) is as tricky as 9a, or I am thinking out the box. It is the claim that "If $F$ is the set of frontier points of $S \subset \mathbb{R}^{n}$, then the set of frontier points of $F$ is $F$.

I have a shortish proof for this: Towards contradiction suppose the set of frontier points of $F$ is not $F$. Then that means $F$ is either the interior of $F$ or $F$ is the exterior points of $F$. Using the interior of $F$, this would mean that we are looking at the interior of the set of frontier points of $S$ which is empty. Which is a contradiction (but then again that would only be the…