"Mathematical Logic" by Cori and Lascar: Incomplete proof of Lemma 1.9?

I have a question on the book "Mathematical Logic: Propositional calculus, Boolean Algebras, predicate calculus" by Rene Cori and Daniel Lascar.

Proof of Lemma 1.9 given on this page is in three parts (bulleted list). Part 2 is where they prove that $\displaystyle o[\neg F] \geq c[\neg F]$ for any propositional formula $\displaystyle F$. $\displaystyle o[\neg F]$ is the number of opening parentheses in $\displaystyle \neg F$ and $\displaystyle c[\neg F]$ is the number of closing parentheses in $\displaystyle \neg F$.

My argument is that this cannot be proven YET for ANY formula $\displaystyle F$, because it hasn't been proven yet for formulas containing parentheses or the symbols $\displaystyle \wedge , \vee , \Rightarrow , \Leftrightarrow$. That is done in part 3. Part 2 proof is only correct for formulas containing propositional variables (since part 1 proves $\displaystyle o[\neg P] \geq c[\neg P]$ for any propositional variable $\displaystyle P$ ) and the symbol $\displaystyle \neg$.

Propositional formulas and propositional variables are defined in this page.

Am I correct or am I missing something?