On page 19, O’Neill comments that the proof of Lemma 33 is a mild generalization of the proof of proposition 28. I think (2) (3) requires spelling out.
Let and let be a co-ordinate system at . Let be a co-ordinate system at . Then by (2) has rank . Thus by exercise 7 and by re-arranging the co-ordinates if necessary, forms a co-ordinate system for on a neighbourhood of .
For the reverse, note that since is a co-ordinate system, by exercise 7, has rank .