The Language $\mathcal{L_0}$: Let $\mathcal{L_0}$ be the smallest set $L$ of finite sequences of $\textit{logical symbols}= \{(\enspace)\enspace\neg\}$ and $\textit{propositional symbols}=\{A_n|n\in\mathbb{N}\}$ for $n \in \mathbb{N}$ satisfying the following properties:
(1) For each propositional symbol $A_n$ with $n\in\mathbb{N}$, \begin{multline} A_n \in L. \end{multline}
(2) For each pair of finite sequences $s$ and $t$, if $s$ and $t$ belong to $L$, then \begin{multline} (\neg s) \in L \end{multline} and \begin{multline} (s \to t) \in L. \end{multline}
Readability for $\mathcal{L_0}$: Suppose that $\phi$ is a formula in $\mathcal{L_0}$. Then exactly one of the following conditions applies.
(1) There is an $n$ such that $\phi = A_n.$
(2) There is a $\psi \in \mathcal{L_0}$ such that $\phi = (\neg\psi)$.
(3) There are $\psi_1$ and $\psi_2$ in $\mathcal{L_0}$ such that $\phi = (\psi_1 \to \psi_2)$
Unique Readability for $\mathcal{L_0}$: Same conditions as Readability, but in (2) and (3), the formulas $\psi$, $\psi_1$, and $\psi_2$ are unique, respectively.
Problem (Polish Notation): Let $\mathcal{P_0}$ be the smallest set of sequences $P$ such that the following conditions hold.
a) For each $n$, $A_n \in P$.
b) If $\psi_1$ and $\psi_2$ belong to $P$, then so do $\neg\psi_1$ and $\to\psi_1\psi_2 = \langle \to \rangle + \psi_1 + \psi_2$.
State and prove the unique readability theorem for $\mathcal{P_0}$