The preservation of cross ratio How to prove that if $a,b,c,d$ and $a',b',c',d'$ are 2 quadruples of distinct points in extended complex plane, and if the cross ratios of these quadruples are equal then there exists Möbius transformation $M$ such that $M(a)=a', M(b)=b',M(c)=c',M(d)=d'$
I am reading a paper and there is already proof that if there exists then the cross ratio is preserved but the remaining is left as an exercise.Any hint would be appreciated.

One definition of cross ratio is particularly apt here: if you begin with  $(z_1,z_2,z_3,z_4)$ and send it by a linear fractional transformation to $(0,1,\infty, u),$ then the cross ratio of $(z_1,z_2,z_3,z_4)$ is defined to be $u.$
After that your exercise is easy.
