"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 for any propositional formula . is the number of opening parentheses in and is the number of closing parentheses in .
My argument is that this cannot be proven YET for ANY formula , because it hasn't been proven yet for formulas containing parentheses or the symbols . That is done in part 3. Part 2 proof is only correct for formulas containing propositional variables (since part 1 proves for any propositional variable ) and the symbol .
Propositional formulas and propositional variables are defined in this page.
Am I correct or am I missing something?