I still havnt solved this and although I have written a half hearted almost certainly incorrect solution I am still interested in knowing if someone else can help me in the right direction.
I started off my halfhearted attempt to a solution by saying that the integral simplifies to
where gamma'(t) is the derivative of a pointwise velocity scaled r'(t)
however the above integral induces the length structure d(x) which by fermats principle should have the same magnitude as the characteristic along which it lies i.e gamma(t) ?? !
So we have proved what we set out to prove.
But in all honesty it looks wrong from any way you look at it. Any suggestions would be welcome.