This is a question I originally posted in Math Stack Exchange, but perhaps the question was too specialized, so I thought I'd post it here instead

I'm currently reading through "Lectures on Quasiconformal Mappings." I'm a bit confused about the proof of Theorem 1 in Chapter 2.

Here's the statement of the theorem and the proof.

I'm actually confused about the Editors' note on the proof of this theorem. It says:

"Shishikura has pointed out to us that the existence of a 'sufficiently fine' subdivision requires proof...

First subdivide Q by both vertical and horizontal lines so that each small rectangle has modulus less than 1/K and any pair of vertically adjacent small rectangles has a neighborhood in which $f$ is K-q.c. The image of each small rectangle then has modulus less than 1, so **one can show by using the Teichmüller extremal problem in Chapter III A that it contains a horizontal line segment**...."

I don't understand how the Teichmüller extremal problem is related to finding the horizontal segments and I don't understand why having the image of the small rectangles have module less than 1 allows one to find horizontal segments.