Definition: Let $X$ be the set of all the symbols on the alphabet of propositional logic, and also let the set $Z$ be the set of all the subsets $Y$ of $X$ such that: if $\alpha$ $\in$ $Y$ then ($\lnot$ $\alpha$) $\in$ $Y$, and if $\alpha$ and $\beta$ belong to $Y$ then ($\alpha$ $\circ$ $\beta$) $\in$ $Y$, such that $\circ$ $\in$ $\{$$\land$, $\lor$, $\to$, $\leftrightarrow$$\}$
We can easily prove the Induction on the Complexity of Formulas as from the definition of formulas above: Let $F$ be the intersection of the family of sets $Z$, then we have that $F$ satisfies the conditions listened above and the is the smallest set (let all the $Y_n$ be all of the elements of $Z$, then it follows that $F$ $\subseteq$ $Y_1$ $\subseteq$ $Y_2$ $\subseteq$ ... $\subseteq$ $Y_n$) such that it belongs to $Z$. As we have a well-founded relation $\subseteq$ on $Z$, then we end the proof by structural induction.
Now, making use of the theorem above, how does one prove the Unique Readibility Theorem? It objectively states that for every formula $A$, one, and only one of the following affirmations are true: (1):= $A$ is atomic, (2):= There is an unique formula $B$ such that $A$ is ($\lnot$$B$), (3):= There are unique formulas $B$ and $C$ such that $A$ is ($B$ $\land$ $C$), or, either, ($B$ $\lor$ C), ($B$ $\to$ $C$), or ($B$ $\leftrightarrow$ $C$).