I tried to prove this theorem using induction: Consider the system $X(t) = F(X(t))$ where F is $C^r$. Then the flow φ(t, X) of this system is a $C^r$ function; that is, $(∂φ/∂t)^r$ and $(∂φ/∂X)^r$ exist and are continuous in t and X.
I know how to show it for $r=1$ but not sure how to generalize it for all $r \in \Bbb{N} $
thanks ahead