Proving that the reversal of a language reversal is that language
Well, we don't really need induction here.

We want to prove that for any language $L$ holds the statement

$$\left( L^R \right)^R = L.$$

(1) Let $s \in \left( L^R \right)^R$.
Then there exists a (unique) string $t \in L^R$, such that $s = t^R$.
The situation is analogous with $t$: there exists a (unique) string $u \in L$ such that $t = u^R$.
From that we get $s = t^R = \left( u^R \right)^R = u$, so $s \in L$. 

(2) Let $s \in L$. Observe that $s^R \in L^R$ and $\left( s^R \right)^R \in \left( L^R \right)^R$, again using the fact that $s = \left( s^R \right)^R$ we get that $s \in \left( L^R \right)^R$.

From (1) and (2) follows the statement of the theorem.