Lemma 3: The map

will move a line which is perpendicular to a line whose slope is m to a vertical line.

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

Clear denominators

m(y1 - y2) = x2 - x1

Remove parentheses

my1 - my2 = x2 - x1

Transpose

x1 + my1 + my2 = x2 + my2

So

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