13. Theorem 2.6.3

Let $f \in M ö b (C)$ , and let $z_{1}, z_{2}, z_{3} \in \hat{C}$ with $w_{1} = f (z_{1})$ , $w_{2} = f (z_{2})$ , and $w_{3} = f (z_{3})$ , and of course $f (z) = w$ . Then we have the following relation: $q (z, z_{1}, z_{2}, z_{3}) =$ 1;; $q (w, w_{1}, w_{2}, w_{3})$ . That is, Möbius transformations preserve cross ratios