It really does not matter what f(x) is, because we choose the endpoints of a parallel secant line which lie on f(x). f(x+n)-f(x-m) / m+n is ALWAYS the gradient of a STRAIGHT LINE. By the above proof, m+n ALWAYS divides f(x+n)-f(x-m).
Unlike Taylor Series which is always an approximation, the Gabriel Polynomial is exact. (On page 5, there is the Gabriel polynomial).