Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space.Let $\ X=(X_1,..,X_n)$ a random vector, with$\ n$ independents random variables whose law is $\mu$ on $\mathbb{R}$. We define $T:\mathbb{R^n}\to\mathbb{R^n}$ such that $T(X_1,..,X_n)=(X_{(1)},..,X_{(n)})$ with $X_{(k)}$ the k-th order statistic.Let $S_n$ set of all permutations $\sigma$ of set $ \left\{ 1,..,n \right\}$ and for any $f:\mathbb{R^n}\to\mathbb{R}$ we define $Sf(x_1...x_n)=\sum_{\sigma \in S^n}f(x_{\sigma(1)},..,x_{\sigma(n)})$.
$\\$ Prove that: $\\$
$\mathbb{E}(f(X)|T(X))=S(f(X))$ a.s.
and if $V(x_1,..,x_n)=(\sum_{i=1}^nx_i,\sum_{i