Let $A$ be a language over $\Sigma$ and let:

$$NE(A) = \setbuilder{w \in A}{w \text{ is not the proper prefix of any string in } A}$$

We show that if $A$ is regular, $NE(A)$ is also regular.

---

Let $M_A = (Q_A, \Sigma, \delta_A, q_{0A}, F_A)$ be a DFA that accepts $A$. We construct a machine $M_N$ that accepts $NE(A)$.

$M_N$ will be identical to $M_A$ except for the final states, so let $M_N = (Q_A, \Sigma, \delta_A, q_{0A}, F_N)$. Define the final states of $M_N$:

$$F_N = \setbuilder{q \in F_A}{\hat{\delta}_A(q, x) \not\in F_A \text{ for any } x}$$

So, $M_N$ is simply a machine that accepts strings that end up in a final state for which there is no additional path to any other final state. Then $M_N$ accepts $NE(A)$, and $NE(A)$ is regular.