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.

Lemma. The group of Möbius transformations $\mathrm{PGL}_2(\Bbb C)$ acts sharply 3-transitively on the Riemann sphere and has three complex dimensions as a manifold. That is, for any pair of 3-tuples of points from the extended complex plane, there is one and only one transformation mapping one point to the other point. I will leave this lemma as a different exercise - it is only linear algebra.
$\bullet$ Let $(z_1,z_2,z_3,z_4)$ and $(w_1,w_2,w_3,w_4)$ be two tuples with the same cross-ratio $r\in\Bbb C$. Since linear equations have unique solutions, $v=w_4$ is the only solution to $[w_1,w_2,w_3;v]=r$. 
So let $A=\bigl(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\bigr)\in\mathrm{PGL}_2(\Bbb C)$ be the unique Möbius transformation that sends $(z_1,z_2,z_3)$ to the point $(w_1,w_2,w_3)$. Applying $A$ to $(z_1,z_2,z_3,z_4)$ and taking the cross-ratio, we have
$$r=[z_1,z_2,z_3,z_4]=[Az_1,Az_2,Az_3,Az_4]=[w_1,w_2,w_3,Az_4]$$
hence $Az_4=w_4$ (by $\bullet$) and $A(z_1,z_2,z_3,z_4)=(w_1,w_2,w_3,w_4)$ as desired. Note this shows that Möbius transformations act sharply transitively on $4$-tuples with a given cross-ratio.
