Sign of a Permutation using Polynomials One way to define sign of permutation is to consider the polynomial 
$P(x_1,x_2,\cdots, x_n)=\prod_{1\leq i<j\leq n} (x_i-x_j)$ and define 
$$Sign(\sigma)= \frac{P(x_{\sigma(1)},x_{\sigma(2)},\cdots, x_{\sigma(n)})}{P(x_1,x_2,\cdots, x_n)}.$$
I want to see the proof of $Sign(\sigma\tau)=Sign(\sigma)Sign(\tau)$. For this, 
$$ Sign(\sigma\tau)=\frac{P(x_{\sigma\tau(1)},x_{\sigma\tau(2)},\cdots, x_{\sigma\tau(n)})}{P(x_1,x_2,\cdots, x_n)}$$ 
$$=\frac{P(x_{\sigma\tau(1)},x_{\sigma\tau(2)},\cdots, x_{\sigma\tau(n)})}{P(x_{\tau(1)},x_{\tau(2)},\cdots, x_{\tau(n)})}\times\frac{P(x_{\tau(1)},x_{\tau(2)},\cdots, x_{\tau(n)})}{P(x_1,x_2,\cdots, x_n)}.$$
I didn't get why the first term on RHS in last step is $Sign(\sigma)$. Can anybody help me? 
(This could be elementary, but I am not justified.)

Hint:
The definition 
$$
Sign(\sigma){}={}\dfrac{P(x_{\sigma(1)},x_{\sigma(2)},\cdots, x_{\sigma(n)})}{P(x_1,x_2,\cdots, x_n)}
$$
is valid for any initial order of $x_1,\ldots,x_n\,$.
For example,
$$
\dfrac{(x_1-x_2)}{(x_2-x_1)}{}={}\dfrac{(x_2-x_1)}{(x_1-x_2)}{}={}-1\,.
$$
