Prove for any natural numbers , we have . So fix and and induct on . For , . Suppose inductively that . We have to prove that . So . Also .
Is this correct?
The answer is “we don’t know”.
I am glad that TPH answered the way he did. I was about do the almost the same thing as he has done. But I would have used different notation. The fact is that there are almost as many notations used in Peano arithmetic as there are textbooks that address the topic. So the reason we don’t is we have not seen the notation you are using.