I have a problem to proof this theorem: "Lex $X$ a vector field in a differentiable manifold $M$ and $f\in\mathcal{C}^\infty(M\to R)$ $\Rightarrow$ for all $k\in\mathbb{N}$ the map $f\circ\Phi_X^t$ is globally defined on the manifold if the flow $\Phi_X^t$ is complete; alternatively the map is defined in an opportune set $D_t(X)$ and $f\circ\Phi_X^t=f+tX(f)+ \dfrac{t^2}{2}X^2(f)+...+ \dfrac{t^k}{k!}X^k(f)+O(t^{k+1})."$
Where $X^k$ is $k$-application of $X$ to $X$.
I have a problem to prove by induction that $\dfrac{d^k}{dt^k}(f\circ\Phi_X^t(p))|_t=(X^k(f(\Phi_X^t))$ in a neighbourhood of $0$.
For $k=1$ the assertion in prooved because
$\dfrac{d}{dt}(f\circ\Phi_X^t(p))|_t=(\Phi_X^t)_{\star,t}(\dfrac{d}{dt}|_t)(f)=(X(f(\Phi_X^t))$,
where the last equality follows from the fact that the flow is the unique solution of the Cauchy's problem $\begin{cases}(\Phi_X^t)_{\star,t}(\frac{d}{dt}|_t)(f)=X(f(\Phi_X^t)) & \\ \Phi_X^0=p \end{cases}.$
Supposed true for $n-1$, when i prove for $n$ I have this:
$\dfrac{d^k}{dt^k}(f\circ\Phi_X^t(p))|_t=\dfrac{d}{dt}(\dfrac{d^{k-1}}{dt^{k-1}}(f\circ\Phi_X^t)|_t)|_t=\dfrac{d}{dt}(X^{k-1}(f(\Phi_X^{t-1})))|_t$ in a neighbour of $0$.
How to complete this? Is possible that I have to use this theorem: "Let $f:M\to N$ a differentiable map between two differentiable manifolds, $X$ a vector field on $M$ and $Y$ a vector field on $N$ such that $Y(f(q))=f_{\star,q}(X(q))$ $\forall q\in M$.
If $f\circ\Phi_X^t$ is an integral curve of $X$ $\Rightarrow f\circ\Phi_X^t $ is an integral curve for $Y$ and in particular $f\circ\Phi_X^t=\Phi_Y^t\circ f."$
If the answer is yes, how can I show that $X^{k-1}(f(\Phi_X^t))$ is an integral curve for $X$? I do not found a book with this generalized Taylor formula to manifold.