Lemma 3.5

  Let T, be as in Lemma 3.4. Let B be the square with vertices . Then for ,

Proof

is the parallelogram with vertices

The cross-section of this with x-coordinate is the vertical interval for . Hence given pick such that and then pick such that . Since and , we see if . Then and by Lemma 3.4 we can pick such that , so Lemma 3.5 follows.