EC: P$14\mathrm{.}10\mathrm{.}10(a,b)[10pts]$ In Problem $5\mathrm{.}5\mathrm{.}1{?}^4$ we defined the quotient of a language $L$ by a letter $a,$ so that the language $L{a}^{-1}$ was defined to be the set $\{w$:wainL$\}.$ We proved there that we could take a regular expression $$ and produce a regular expression for $L()a^{-1}.($a$)$ The quotient of a language $L$ by a string $w$ is the set $L{w}^{-1}=\{u$:uwinL$\}.$ Argue using the result of Problem $5\mathrm{.}5\mathrm{.}1$ that $L{w}^{-1}$ must be regular if $L$ is