Proof of Möbius transformation cross-ratio preservation theorem

From the definition of Möbius transform we getthat the image $w_{k}$ of a point $z_{k}$ is

w_{k}=\\frac{az_{k}+b}{cz_{k}+d}

From this we get

w_{i}-w_{j}=\\frac{az_{i}+b}{cz_{i}+d}-\\frac{az_{j}+b}{cz_{j}+d}=\\frac{(ad-bc)(%z_{i}-z_{j})}{(cz_{i}+d)(cz_{j}+d)}

and by inserting this into the cross-ratios

\\frac{(w_{1}-w_{2})(w_{3}-w_{4})}{(w_{1}-w_{4})(w_{3}-w_{2})}=\\frac{\\frac{(ad-%bc)(z_{1}-z_{2})}{(cz_{1}+d)(cz_{2}+d)}\\frac{(ad-bc)(z_{3}-z_{4})}{(cz_{3}+d)(%cz_{4}+d)}}{\\frac{(ad-bc)(z_{1}-z_{4})}{(cz_{1}+d)(cz_{4}+d)}\\frac{(ad-bc)(z_{%3}-z_{2})}{(cz_{3}+d)(cz_{2}+d)}}=\\frac{(z_{1}-z_{2})(z_{3}-z_{4})}{(z_{1}-z_{%4})(z_{3}-z_{2})}