proof of converse of Möbius transformation cross-ratio preservation theorem
Suppose that are distinct. Consider the transform defined as
Simple calculation reveals that , , and .Furthermore, equals the cross-ratio of .
Suppose we have two tetrads with a common cross-ratio . Then, as above, we mayconstruct a transform which maps the first tetrad to and atransform which maps the first tetrad to . Then maps the former tetrad to the latter and, by the group property, it is also aMöbius transformation.