I am reading a differential equation book that discusses the Brachistochrone problem. The book discusses Bernoulli's solution that uses Snell's law. The book says that a ray would follow the fastest path from point A to point B. Then it proves Snell's law for a light ray that starts from point A and then goes into a denser medium and finally reaches point B.   My question is, how can I prove that Snell's law would still provide the fastest path if the light goes through N layers of the medium before reaching its final destination?

I am reading a differential equation book that discusses the Brachistochrone problem. The book discusses Bernoulli's solution that uses Snell's law. The book says that a ray would follow the fastest path from point A to point B. Then it proves Snell's law for a light ray that starts from point A and then goes into a denser medium and finally reaches point B. 

My question is, how can I prove that Snell's law would still provide the fastest path if the light goes through $N$ layers of the medium before reaching its final destination?