Will this suffice?.

Pocklington's Theorem: Let n-1 = qkR where q is a prime which does not divide R. If there is an integer a such that an-1 = 1 (mod n) and gcd(a(n-1)/q-1,n) = 1, then each prime factor q of n has the form qkr+1.

Proof. Let p be any prime divisor of n, and let m be the order of a modulo p. As above m divides n-1 (first condition on a), but not (n-1)/q (second condition); so qk divides m. Of course m divides p-1 so the conclusion follows.