For a language $L$ over an alphabet $\Sigma$, define the language $\operatorname{HasPrefix}(L)$ as follows: $$\operatorname{HasPrefix}(L) = \{xy\mid x,y \in\Sigma^*, x \in L, xy \in L, |y|> 0\}$$ Note the last condition, which requires that y be nonempty.
I want to prove that if $L$ is regular, then so is $\operatorname{HasPrefix}(L)$.
Any suggestions on how to approach this? I want to prove it by making DFAs for the language L and HasPrefix(L) to show that they're both regular.