[compl.ana]

*Cross-ratio as a complete invariant on ordered 4-tuples*:

Let be 4-tuples of distinct points in , and define them to be equivalent if there is a Mobius transformation sending one to the other.

Then and are equivalent if and only if the cross-ratios are the same.

Because*:
*. If are equivalent then .

. Let . Then sends . Furthermore since , we have that .