Running time for Testing Polygonal Neighbours for Intersection or Inclusion
In the paper you reference, Theorem 5 is a claim about simple polygons, where Theorem 6 deals with convex polygons, which are a special kind of simple polygons.

The 'As we have seen before, ...' refers to this line: 'As we have seen before, each such test can be done in $O(k)$ time.$^8$', where $8$ references another paper.

However, even though I do not know why you are reading this paper, I'd advise against it. This paper is pretty old and the main result (the $n\log n$ segment intersection algorithm) has been both improved upon and better explained by other sources. If you did not have an introductory course on computational geometry or something similar, I advise you to look at a computational geometry /algorithmic geometry textbook or lecture notes first.