Möbius transformation cross-ratio preservation theorem

A Möbius transformation $f:z\\mapsto w$ preserves the cross-ratios, i.e.

\\frac{(w_{1}-w_{2})(w_{3}-w_{4})}{(w_{1}-w_{4})(w_{3}-w_{2})}=\\frac{(z_{1}-z_{%2})(z_{3}-z_{4})}{(z_{1}-z_{4})(z_{3}-z_{2})}

Conversely, given two quadruplets which have the same cross-ratio, thereexists a Möbius transformation which maps one quadruplet to the other.

A consequence of this result is that the cross-ratio of $(a,b,c,d)$ is thevalue at $a$ of the Möbius transformation that takes $b$ , $c$ , $d$ , to $1$ , $0$ , $\\infty$ respectively.