Summation of Products of n Numbers taken m at a time with Repetitions/Lemma 2

Theorem
Let $a, b \in \Z$ be integers such that $b \ge a$.

Let $U$ be a set of $n = b - a + 1$ numbers $\set {x_a, x_{a + 1}, \ldots, x_b}$.

Let $m \in \Z_{>0}$ be a (strictly) positive integer.

Let:

That is, $h_m$ is the product of all $m$-tuples of elements of $U$ taken $m$ at a time.

For $r \in \Z_{> 0}$, let:
 * $S_r = \ds \sum_{j \mathop = a}^b {x_j}^r$

Let $\map G z$ be the generating function for the sequence $\sequence {h_m}$.

Then: