I can't understand what you think you can see here: , since, for example, .
Anyway, and since , for simplicity of the proof we'll prove that in fact
Now, we can form pairs , so
, so we have to prove this last sum is divisible by p (the minus sign never minds: we can take it out of the sum).
But as the only elements in which are inverses to themselves are 1 and -1 and without these two we're left with an even number of
elements, from 2 to p-2 (and here enters the assumption ), where we can pair each one with its inverse. Now, the well-known formula
means the sum equals zero modulo p since, again, p is prime greater than 5. Q.E.D.