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- #1

- Mar 10, 2012

- 834

\begin{align*}

\bigcap_{i=1}^{r}(\sigma_i\circ\varphi_i)^{-1}(j)&=\bigcap_{i=1}^{r}(\tau_i\circ\varphi_i)^{-1}(l)\neq\emptyset\\

\Rightarrow(\sigma_i\circ\varphi_i)^{-1}(j)&=(\tau_i\circ\varphi_i)^{-1}(l),\forall i\in\{1,\ldots,r\}~~~(\text{from prop. 9})\\

\Rightarrow\varphi_i^{-1}(\sigma_i^{-1}(j))&=\varphi_i^{-1}(\tau_i^{-1}(l))\\

\Rightarrow\sigma_i^{-1}(j)&=\tau_i^{-1}(l)

\end{align*}

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