Sign_of_Permutation_is_Plus_or_Minus_Unity_v2

Let $n \in \N$ be a natural number. Let $\N_n$ denote the set of natural numbers $\set {1, 2, \ldots, n}$. Let $S_n$ denote the symmetric group on $n$ letters. Let $\sequence {x_k}_{k \mathop \in \N_n}$ be a finite sequence in $\R$. Let $\pi \in S_n$. Let $\map {\Delta_n} {x_1, x_2, \ldots, x_n}$ be the product of differences of $\tuple {x_1, x_2, \ldots, x_n}$. Let $\map \sgn \pi$ be the sign of $\pi$. Let $\pi \cdot \map {\Delta_n} {x_1, x_2, \ldots, x_n}$ be defined as $\pi \cdot \map {\Delta_n} {x_1, x_2, \ldots, x_n} := \map {\Delta_n} {x_{\map \pi 1}, x_{\map \pi 2}, \ldots, x_{\map \pi n} }$ Then either $\pi \cdot \Delta_n = \Delta_n$ or $\pi \cdot \Delta_n = -\Delta_n$ That is $\map \sgn \pi = \begin{cases} 1 & :\pi \cdot \Delta_n = \Delta_n \\ -1 & : \pi \cdot \Delta_n = -\Delta_n \end{cases}$ Thus $\pi \cdot \Delta_n = \map \sgn \pi \Delta_n$