$\newcommand{\Ga}{\Gamma}$

I am trying to find a proof of Green's formula for rectifiable Jordan curves $\Ga$ (and the corresponding interior regions $R$). There is a proof by Ridder, followed by very similar proofs by Verblunsky and Potts. The idea in those three papers, which looks quite natural, is to subdivide $R\cup \Ga$ into small regions $R_i\cup \Ga_i$ by vertical and horizontal lines. The problem is how to deal with the "partial" regions $R_i\cup \Ga_i$ that are not rectangles; more precisely, with the line integrals over the corresponding $\Ga_i$'s. To deal with such line integrals rigorously, one first of all needs to parametrize the boundaries $\Ga_i$. However, none of the mentioned three authors (or any other ones known to me) even mentions such a parameterization.

In Mathematical Analysis: a Modern Approach to Advanced Calculus, 1957, by Apostol, an apparent attempt is made to make Ridder's approach rigorous. The Green formula in question is stated there as Theorem 10--43. The crucial part in the proof of Apostol's Theorem 10--43 is Theorem 10--42, which tries to deal with the mentioned "partial" regions $R_i\cup \Ga_i$. However, there are a few places in the proof of Apostol's Theorem 10--42 that I don't understand.

Let $\Ga$ be a rectifiable Jordan curve bounding the corresponding interior region $R$. First, Apostol takes a lowest and highest points, $p_1$ and $p_2$ on $\Ga$. These two points are then separated by a horizontal line, say $\ell$. Then $p_1$ and $p_2$ are joined by "arcs" $C_1$ and $C_2$ to points $q_1$ and $q_2$ in $R$ close to $p_1$ and $p_2$, respectively, so that $q_1$ is below $\ell$ and $q_2$ is above $\ell$.

Question 1.Apostol says such "arcs" (possibly non-rectifiable) exist, without proof. I do not immediately see how to prove this, even though it's probably simple. (The $C_j$'s are never mentioned again in the proof, after they are introduced.)

Then the points $q_1$ and $q_2$ are connected by a polygonal line $C\subset R$, which must have an intersection point with $\ell$. Without loss of generality, there are finitely many such points. For each such point $p$, let $[u,v]$ be the maximal segment on $\ell$ such that the open interval $(u,v)$ is contained in $R$. Then Apostol says there must be at least one such maximal segment $[u,v]$ that has an odd number of intersections with the polygonal line $C$; I guess this follows because the total number of the intersections of $C$ with $\ell$ can be proved to be odd, by induction. But then Apostol just says: "Using such a segment, it is clear that $L[u,v]$[$=[u,v]$ -- I.P.] forms, with $\Ga$, two rectifiable Jordan curves $\Ga_1$ and $\Ga_2$, one of which contains $p_1$ (call this one $\Ga_1$) and the other contains $p_2$. Moreover, these curves form the boundaries of two regions $R_1$ and $R_2$, whose union is $R$. The positively oriented boundaries $\Ga(R_1)$ and $\Ga(R_2)$ of $R_1$ and $R_2$ such that $\Ga(R)=\Ga(R_1)+\Ga(R_2)$."

There is a picture there, making the above claims plausible. Yet, here is my main question:

Question 2.Nothing in that passage is clear to me. How does this "forming" of curves $\Ga_1$ and $\Ga_2$ occur? Are they Jordan curves? If so, why? How are $\Ga_1$ and $\Ga_2$ parameterized? Why is the union of $R_1$ and $R_2$ equal to $R$? What precisely are the roles of the points $q_j$ and the "arcs" $C_j$ in this proof?

Can one help me decipher this proof? Or, perhaps even better, is there a completely rigorous, fully detailed proof of this obviously important and natural result? (Of course, there is a general Stokes theorem for manifolds, but it does not seem at all obvious how to build a bridge from there to Jordan curves.)