.  [*Example 0.*]{} Consider a fuzzy automaton $\widetilde{G}=(\widetilde{Q}_{0},\widetilde{E},\widetilde{\delta}, \widetilde{q}_{0})$, where $\widetilde{E}=\{\widetilde{a},\widetilde{b}, \widetilde{c}\}$, $\widetilde{q}_{0}$=$[0.0, 0]$, and $$\widetilde{a}=\left[ \begin{array}{cc} 0.0 &\ 0.0\\ 0&\ 0.0 \end{array} \right], \widetilde{b}=\left[ \begin{array}{cc} 0.0 &\ 0.0\\ 0&\ 0.0 \end{array} \right], \widetilde{c}=\left[ \begin{array}{cc} 0.0&\ 0\\ 0.0&\ 0.0 \end{array} \right].$$ Let $pr(\widetilde{K})$ be generated by a fuzzy automaton $\widetilde{H}=(\widetilde{Q}_{0},\widetilde{E},\widetilde{\delta},\widetilde{p}_{0})$, where $\widetilde{p}_{0}$=$[0.0,0]$, $\widetilde{E}=\{\widetilde{a},\widetilde{b}, \widetilde{c}\}$, and $\widetilde{a},\widetilde{b}$ are the same as those in $\widetilde{G}$ , but $\widetilde{c}$ is changed as follows: $$\widetilde{c}=\left[ \begin{array}{cc} 0.0&\ 0\\ 0.0&\ 0.0 \end{array} \right].$$ Suppose that $\widetilde{\Sigma}_{uc}$ and $\widetilde{\Sigma}_{o}$ are defined as