Where can I find the proof of this theorem about the smoothness of the initial data? What to follow is a reference from the text Differential forms by Victor Guillemin and Peter Haine

So clearly the theorems $2.2.4$ and $2.2.5$ follow directely by the Cauchy theorem for ordinary differential equation provided that the vector field if locally lipschitz or of class $C^r$ for $r\ge 1$ since any function of calss $C^1$ is locally lipschitz. However how prove the last theorem? In particular I know that if the vector field $\vec v$ is of class $C^r$ for $r\ge 1$ then the solution of the correspondent system of differential equations is of class $C^r$ too but unfortunately this did not help me to prove the theorem. So could someone idicates where I can find the proof of the last theorem, please? I point out I did NOT study lebesgue integration and measure theory so that I ask courteously to not use them, thanks.
MY PROOF ATTEMPT
So by the regularity theorem we know that the function $\gamma_p$ is of class $C^\infty$ so that if $A_p$ is an open neighborhood of any $p\in V$ contained in $V$ then the set $J:=\gamma^{-1}_p[A_p]$ is open an open interval containing $a\in I$ and so if $\pi$ is the projection of $V\times I$ onto $I$ the statement follows proving that
$$
h(q,t)=[\gamma_p\circ\pi](q,t)
$$
for any $(q,t)\in A_p\times J$ because the set $A_p\times J$ is open and the function $\gamma_p\circ\pi$ is of class $C^\infty$ being a composition of such functions. So how prove the last equality? Is effectively it hold?
There are quite a few alternative proofs of Theorem 2.2.6 and of many of its variants.
Incidentally, Theorem 2.2.6 is just a very particular case of the general situation (most notably, in general it is quite restrictive to assume that $I$ is the same for all $p$ as in Theorem 2.2.6 but it is fine in case we just care about a small neighborhood).
Unfortunately, there are no simple proofs (read short if you want since all proofs are classical), although in the present case most tend to be slightly simple since it is much harder to prove that $C^k$ implies $C^k$ for some finite $k$ then for $k=\infty$. The reason is that for a finite $k$ it is usually necessary a different argument to obtain say $C^k$ and not only $C^{k-1}$ (I am really simplifying this a lot since the theory is quite vast). Still all these proofs for $k=\infty$ are quite long.
Overall it is thus better to give a pointer to a canonical reference, such as Ordinary Differential Equations, by Jack K. Hale that unfortunately is not with us anymore. Have a look at Theorem 3.3 in page 21 (a bit too concise but one of the best places).
There are quite a few alternative proofs of Theorem 2.2.6 and of many of its variants.
Incidentally, Theorem 2.2.6 is just a very particular case of the general situation (most notably, in general it is quite restrictive to assume that $I$ is the same for all $p$ as in Theorem 2.2.6 but it is fine in case we just care about a small neighborhood).
Unfortunately, there are no simple proofs (read short if you want since all proofs are classical), although in the present case most tend to be slightly simple since it is much harder to prove that $C^k$ implies $C^k$ for some finite $k$ then for $k=\infty$. The reason is that for a finite $k$ it is usually necessary a different argument to obtain say $C^k$ and not only $C^{k-1}$ (I am really simplifying this a lot since the theory is quite vast). Still all these proofs for $k=\infty$ are quite long.
Overall it is thus better to give a pointer to a canonical reference, such as Ordinary Differential Equations, by Jack K. Hale that unfortunately is not with us anymore. Have a look at Theorem 3.3 in page 21 (a bit too concise but one of the best places).