Question about the TypeIII region in Green's Theorem

Question

I'm trying to understand the proof of Green's Theorem. Every proof I see so far goes like the following: we consider Type I region then Type II region (described here https://en.wikipedia.org/wiki/Green%27s_theorem) and then the theorem holds for Type III which is a region that happens to be both Type I and Type II.

I have two questions:

1. I am confused with what Type III are... If type I means left and right curves are vertical lines and type II means top and bottom curves are horizontal lines then doesn't that mean Type III region is a rectangle?

2. The final part of the proof is that the interior of the piceswise smooth simple closed curve can be decomposed into (finite?) number of Type III regions. Can someone provide a reference for this fact?

Thank you!