Let $A$ be any language. Define $\text{DROP-OUT(A)}$ to be the language containing all strings that can be obtained by removing one symbol from a string in $A.$ Thus, $\text{DROP-OUT(A) = $\{xz| xyz\in A$ where $x, z\in\sum^{*},y\in\sum$\}}.$ Show that the class of regular languages is closed under the $\text{DROP-OUT}$ operation. Give both a proof by picture and a more formal proof by construction as in $\text{Theorem 1.47.}$
