Let $L$[0]$$ be defined formally as $$\backslash \{$x$\_10$x$\_2..\mathrm{.}0$x$\_$n $|$ x$\_$i $\backslash$in $\backslash$Sigma$,$ x$\_1...$x$\_$n $\backslash$in L$\backslash \}$$$.$ In other words, strings of $L$[0]$$ are formed by taking a string from $L$ and inserting a $$0$$ between each character of the word, so that if $L $=\backslash \{\backslash$bl$\{11\},\backslash$bl$\{1010\}\backslash \}$$$,$ then $L$[0]=\backslash \{\backslash$bl$\{1\}\backslash$rd$\{0\}\backslash$bl$\{1\},\backslash$bl$\{1\}\backslash$rd$\{0\}\backslash$bl$\{0\}\backslash$rd$\{0\}\backslash$bl$\{1\}\backslash$rd$\{0\}\backslash$bl$\{0\}\backslash \}$$$.$ Prove that if $L$ is regular, so is $L$[0]$$$.$