A difficulty in understanding theorem 10.6.7 in Petrovic.(n-dimensional intermediate value theorem) The theorem and its proof are given below:

And those are the used theorems:


But I have a difficulty in understanding the last 2 lines in the proof of the theorem 10.6.7., My questions are:
1- By representing $M$ by this form, what is $M$? is it a line, a point? should parentheses be added? and why this leads that $f(M) = 0.$ 
Could anyone explain to me those points please?     
 A: $M$ is an $n$-dimensional point. In particular, as stated, $M \in A \subset \mathbb{R}^n$. I agree that parenthesis should've been added to make this clear. I'm not sure why the book didn't do this, unless perhaps it was just an oversight or they believed this was clear from the context.
As for how the proof leads to concluding that $f(M) = 0$, I'm not sure what point(s) you're not clear on so I will paraphrase the whole proof.  It uses that, by the definition of a polygonally connected domain, there exists a finite set of interconnected straight line segments between the initial point of $P$ and the final point of $Q$, all fully contained within $A$. Starting at $P_0 = P$, consider the value at $f(P_1)$ is one of $\lt 0$, $= 0$ or $\gt 0$. If equal to $0$, then this is the $M$ you're looking for. If it's $\lt 0$, then continue to the next point. If not here, then eventually you will get a line segment where the end point is either $0$ or $\gt 0$ because, for the final point, we have $f(Q) \gt 0$. If the case is there is a line segment where $f$ is $\lt 0$ at the start and $\gt 0$ at the end, then parametrize this line segment using the equation in $t$ given in $10.11$. Then using its Theorem $10.3.6$, the given function $F$ defined in $10.12$ is continuous for $0 \lt t \lt 1$. As $F(0) \lt 0$ and $F(1) \gt 0$, by its Theorem $3.9.1$, using continuity, there must be a value of $t$ where $F(t) = 0$. As stated, let $M$ be the point determined by this value of $t$.
In summary, the proof shows there's at least one point $M$ which is either at one of the straight line vertices or is determined by the parametric equation in $t$, with this $M$ being such that $f(M) = 0$.
I hope this answers your question and makes the theorem proof clear to you. Please let me know if there's anything you still don't understand.
