Q$3(5$ points$)$

Question $3-$Regular languages closed under operations

Let $$ be some alphabet. For a natural number $k,$ we define the following operation on

the set of all languages over $.$ The operation MOVE $-k(L_1,L_2)$ takes in two languages

$L_1$ and $L_2$ and outputs the language $L_2$ where the set of words of length $k$ from $L_2$

have been replaces with the set of words from $L_1$ with length $k$ :

MOVE$-k(L_1,L_2)=\{win{}^{**}||w|=k$ and $win{L}_1,or|w|k$ and $win{L}_2\},$

where ${}^{**}$ is the set of all words over $.$

Prove that, for any kinN the class of regular languages is closed under the operation

MOVE$-k.$