How to prove that the reversal of the concatenation of two strings is the concatenation of the reversals?
Well, consider the following "proof" (I'll explain afterwords why the quotation marks):

Let $w=a_1\cdots a_n$ be a word, then we have that

$w\in (L_1L_2)^{\mathrm{rev}}$ iff 

$a_n\cdots a_1\in L_1L_2$, iff 

there exists $0\le k\le n$ such that $a_n\cdots a_k\in L_1$ and $a_{k-1}\cdots a_1\in L_2$, iff 

there exists $0\le k\le n$ such that $a_k\cdots a_n\in L_1^{\mathrm{rev}}$ and $a_{1}\cdots a_
{k-1}\in L_2^{\mathrm{rev}}$, iff 

$a_1\cdots a_n\in L_2^{\mathrm{rev}}L_1^{\mathrm{rev}}$ iff

$w\in L_2^{\mathrm{rev}}L_1^{\mathrm{rev}}$

In my opinion, this proof is correct, formal, and would be accepted as a proof of the claim.

However, each "iff" along the way (apart from the last one) actually hides an inductive proof. The point is that each of these inductions is extremely trivial.
In fact, even writing $w=a_1\cdots a_n$ actually hides some induction. So you need to ask yourself how "deep" do you want to go into the technicalities. 

If you want to prove this claim from basic principles, i.e. from the axioms of set theory, then you need to formalize everything. This would be very painful, but such things are actually useful when we use computer-assisted proofs.