The Horizontal Lemma

This chapter contains the statement and proof of the Horizontal Lemma. The proof follows a similar outline as in Chapter 10. The chapter is organized as follows. Section 11.2 uses symmetry to reduce the Horizontal Lemma to a simpler statement. Section 11.3 modifies the construction as in the vertical case, reducing the Horizontal Lemma to the simpler Lemma 11.4. Section 11.4 proves two easy technical lemmas. Section 11.5 proves the Λ‎' version of Lemma 11.4. Section 11.6 keeps track of the signs and prove Lemma 11.4 as stated.