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See at p. 163 of this PDF link:

https://eclass.uoa.gr/modules/document/file.php/PHYS289/Βιβλία/Arnold%2C%20V.I.%20-%20Ordinary%20differential%20equations_Red.pdf

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).

See at p. 163 of this PDF link:

https://eclass.uoa.gr/modules/document/file.php/PHYS289/Βιβλία/Arnold%2C%20V.I.%20-%20Ordinary%20differential%20equations_Red.pdf

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 the flaw is explained right after it. Then the proof of the proposition you want starts (p.164).

See at p. 163 of this PDF link:

https://eclass.uoa.gr/modules/document/file.php/PHYS289/Βιβλία/Arnold%2C%20V.I.%20-%20Ordinary%20differential%20equations_Red.pdf

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 (but not corrected; that is left as a problem for the reader). Then the proof of the proposition you want starts (p.164).

Source Link

See at p. 163 of this PDF link:

https://eclass.uoa.gr/modules/document/file.php/PHYS289/Βιβλία/Arnold%2C%20V.I.%20-%20Ordinary%20differential%20equations_Red.pdf

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 the flaw is explained right after it. Then the proof of the proposition you want starts (p.164).