My question is how to prove that $\Sigma$-formulas are uniquely readable (in our course this was wasn't really proved - in the proof it said just "proof by induction", but I'm confused what was meant by that, this we aren't dealing with numbers on which I can induct; was an induction by the length of the formula meant ? A colleague said,, that probably induction on the construction of the formula was meant, but I don't know what this menas either). I would very much like to see a proof, so that I can get the idea and use this type of proof later on my own, since a lot of "proofs by induction" which involve formulas follow...
Sadly, for this I have to introduce some definitions since they seem to vary from book to book and I haven't found a book, which uses similar definitions to the ones in your course (and since my questions are very much about the details, I can't use a different seet of defintions). So after these apologies, here comes the definition:
We have defined the set $\Sigma_{Term}$ of $\Sigma$-formulas as the smallest set of strings over the alphabet $\Sigma\cup\mathbb{V}$ (where $\Sigma$ is the signature and $\mathbb{V}=\left\{v_0,v_1,v_2,\ldots \right\}$ is a countable set of variables) such that:
$ \blacktriangleright$ every variable is a term
$ \blacktriangleright \ \ \ \mathtt{ft_1\ldots t_n}\in \Sigma_{Term}$ if $\mathtt{t_1,\ldots ,t_n}$ were $\Sigma$-terms and $\mathtt{f}$ was a symbol for a function of arity $n$ in the signature $\Sigma$