is the class of languages that can be decided in polynomial time by a non-deterministic Turing Machine. Now let be the class of languages that can be recognized in polynomial time by a non-deterministic Turing machine.
We can ask the same question: Is ?
The proof is almost the same, except that we use non-deterministic Turing Machine, instead of deterministic one. For completeness of the detail, I will write the proof.
Proof. Clearly, . Now, let . Then there exists a non-deterministic Turing Machine that will accept in time, for some polynomial . Now, we construct a non-deterministic Turing Machine that will decide in polynomial time. Let on input runs a simulation of in time. If it halts and ACCEPT, then also accepts. Otherwise, REJECT.
Still, the proof is non-constructive.