Let be a language over and let:
We show that if is regular, is also regular.
Let be a DFA that accepts . We construct a machine that accepts .
F_N = \setbuilder{q \in F_A}{\hat{\delta}_A(q, x) \not\in F_A \text{ for any } x}
$$
So, 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 accepts , and is regular.