I am currently reading the proof of the connectivity theorem of Wilberd van der Kallen (Theorem 2.6 in https://link.springer.com/article/10.1007/BF01390018) for a seminar talk. I am a little stuck on the details of section 2.15. I understand the basic inductive process but I am not sure why $\mathcal{O}(X)\cap F$ is $d$-connected by the induction hypothesis. I can see that the set is very similar to $\mathcal{O}(R^{n-1}+e_{n}\delta)$ which should be $d$-connected by induction but they are not the same as far as I can tell. I think I might be misintepreting the induction hypothesis. Furthermore I am unsure how to check the conditions for Lemma 2.13.

Somewhat unrelated I was wondering why he chooses the "size" $2n$ for the poset $\mathcal{O}(R^n+e_{n+1}\delta)\cap\mathcal{U}$ in section 2.14. I guess we are choosing a size for the induction to work properly but why $2n$ and not just $n$?

Any help is appreciated!