I am re-reading the proof of Ehrenfeucht's theorem on page 90 of Mathematical Logic of Shoenfield. I have the following problem. For the sake of clarity I have highlighted in red the problematic part in the theorem. The question is: How is it possible infer by the lemma 3 that the proof uses special axioms only for $\mathbf r_1$, ... , $\mathbf r_n$ and the special constants in $\mathbf A_1$, ... , $\mathbf A_{k-1}$? (For clarity reasons I have highlighted too in the lemma the part that I consider being referred to by the theorem.) Finally, for proof ends, it is to be noted that this fact is redundant. Thanks in advance for any suggestion.
I was wrong, the argument is correct. I give the answer in case someone were to have my own doubts. First, the part I stressed is essential to the proof for the generality of $\mathbf A$ (i.e., the $\mathbf C'$ must be the same for every $\mathbf A$). Second (I found this irksome), the fact that at most there are special axioms for those special constants, derives by interpreting lemma 3 in such a way that $T_c$ is a conservative extension of $T^*=T \cup \{r_1,...,r_m,$ special axioms of these special constants$\}$ and considering $T'$ starting by $T^*$ by adding the remaining special constants (i.e., we can use the same proof of the lemma adapting it to this case).