See at p. 163 of this PDF link:
Maybe you got confused because between the statement and the proof of the proposition you are interested in, there is a technical Lemma with its "proof". Moreover, this "proof" is willingly flawed (plain Arnold's style!), and just after the flaw is explained right after it(but not corrected; that is left as a problem for the reader). Then the proof of the proposition you want starts (p.164).