Let's consider the following:
Lemma Let $g\in F(\beta)$ and $f\in \Bbb R^\Bbb R$ such that $f\restriction M= g\restriction M$ for some $M\subset \mathbb R$. Then $f\in F(\beta).$
Of course, no one will be able to follow the question without saying what $F(\beta)$ is. In my case, $F(\beta)$ needs to be constructed by transfinite induction or another way. I know the most natural way is that construct $F(\beta)$ and then state the lemma. But for some, this was hard for me. This is why I am looking for another way to do this. Here I have some questions:
Q1 Is true if I state the lemma, then give the construction of $F(\beta)$, and provide the proof of the lemma? If so, If this kind of idea sometimes use in book or journal paper?
Q2 If not, do you recommend an idea to play around this without introducing $F(\beta)$ first?
Thank you in advance.