Let [tex] \rm x_1,x_2, \dots,x_n \in \mathbb{R}[/tex] be distinct. Prove that

[tex] \rm\sum \limits_{1 \leq i \leq n} \: \prod \limits_{j \ne i} \frac{1 - x_{i} x_{j}}{x_{i} - x_j} = \rm \begin{cases}0 \rm, \: if \: n \: is \: even; \\ 1, \: \rm if \: n \: is \:o dd.\end{cases}\\ [/tex]
​