Equality involving sums over permutations. Im reading The Strong Law of Large Numbers for U-statistics by Wassily Hoeffding (1961). 
Let $h:\mathcal{X}^m \to \mathbb{R}$ be any measurable mapping (i.e. in general non-permutation-symmetric in its arguments). It is stated that since the arithmetric mean mapping $g$ defined by
$$
(X_1,...,X_n) \stackrel{g}{\mapsto} \frac{(n-m)!}{n!}\sum_{1\leq i_1 \not = \cdots \not = i_m\leq n} h(X_{i_1},...,X_{i_m})
$$
is permutation-symmetric (in its arguments), each term may be placed by the arithmetric mean of the m! terms whose supscripts are permuations of the same set of integers. That is
$$
\frac{(n-m)!}{n!}\sum_{1\leq i_1 \not = \cdots \not = i_m\leq n} h(X_{i_1},...,X_{i_m})$$
$$
 =\frac{(n-m)!}{n!}\sum_{1\leq i_1 \not = \cdots \not = i_m\leq n} \frac{1}{m!} \sum_{\sigma\in \Pi_m}h(X_{i_{\sigma(1)}},...,X_{i_{\sigma(m)}}), 
$$
where $\Pi_m$ is the set of all permutations of $\{1,...,m\}$.
Could anyone help me with this equality, I can't seem to tackle it?

Fix an ordered sequence $(i_1, \ldots, i_m)$ with $i_j \in [n]$ all different. Let us define the sum over all permutations $\sigma \in \Pi_m$ by
\begin{equation}
S(i_1, \ldots, i_m) = \sum_{\sigma \in \Pi_m} h(X_{\sigma(i_1)}, \ldots, X_{\sigma(i_m)}).
\end{equation}
The sum in $S(i_1, \ldots, i_m)$ contains $h(X_{i_1}, \ldots, X_{i_m})$ as a term but also many other terms. Now, sum over all possibilities for $(i_1, \ldots, i_m)$ obtaining
\begin{equation}
\sum_{(i_1, \ldots, i_m)} S(i_1, \ldots, i_m)
\end{equation}
Notice that for a fixed $(j_1, \ldots, j_m)$ the term $h(X_{j_1}, \ldots, X_{j_m})$ will appear many times in this sum. More precisely, it will appear once for every $(i_1, \ldots, i_m)$ that is a permutation of $(j_1, \ldots, j_m)$ and there are $m!$ of them. Thus, we have
\begin{equation}
\frac{1}{m!} \sum_{(i_1, \ldots, i_m)} S(i_1, \ldots, i_m) = \sum_{(j_1, \ldots, j_m)} h(X_{j_1}, \ldots, X_{j_m})
\end{equation} 
