problem extending a function on the tangent bundle I have problem with a extension of a function in Poor's book page 48. Let $M$ be a smooth manifold and $p \in M$. Let $h:T_pM \longrightarrow M$ smooth function that $h(0_p)=p$ and exists $V\subset T_pM$ open neighbourhood of $0_p$, $U\subset M$ open neighbourhood of $p$ with $h|_V:V\longrightarrow U$ diffeomorphism.
Define $f:TU \longrightarrow M$, $f(w)=h(h^{-1}(\pi(w))+\text{pr}_2(dh_{\pi(w)}^{-1}(w)))$. Where $\text{pr}_2:TV\longrightarrow V$ is the "velocity projection" and $\pi:TM \longrightarrow M$ is the bundle projection.
If $w \in V$, then $\pi(w)=p$, $f(w)=h(\text{pr}_2(dh_{p}^{-1}(w)))$. The book says $f(w)=h(w)$ but I don't see why.
Why $f|_V=h|_V$? I think, probably the author forgot some hypothesis about $h$...

Probably the condition the author forgot is that the derivate at $0_p$, $dh_{0_p}:T_pM\longrightarrow T_pM$ is the identity. As the book only uses $h=x^{-1}\circ dx_p$. Then $f(w)=h(w) \;\; \forall w \in V$.
