Lemma 2: The map

will move a line whose slope is m to a horizontal line.

Proof: Let (x1, y1) and (x2, y2) be points on a line whose slope is m. Then

Clear denominators

m(x1 - x2) = y1 - y2

Remove parentheses

mx1 - mx2 = y1 - y2

Transpose

y1 - mx1 = y2 - mx2

So

and the images of the two points have the same y-coordinate.