I found the following question with an answer here, but I can’t understand the steps of the solution.

Show that if a language $A$ is in RE and $A leq_m overline{A}$, then $A$ is recursive.

Solution.Since $A leq_m overline{A}$, it follows that $overline{A} leq_m A$, and since $A$ is in RE, it follows that $overline{A}$ is also in RE. Since both $A$ and $overline{A}$ are in RE, it follows that $A$ is in R (this follows from a theorem you learned in class).

Here $le_m$ demotes mapping reducibility.

Actually I can’t understand most of the answer. In particular:

- Why does $A leq_m overline{A}$ imply $overline{A} leq_m A$?
- I understand the following step (why $overline{A}$ is in RE).
- Which theorem is used to deduce that $A$ is in R? (I’m not a student in this class)