k-symbol differential opeartor L, and its independent of choices.

This material is in O. Well's Differential analysis on complex manifold, page 115.

Let, $(x,v) \in T'(X)$ ($T^*(X)$ with deleted zero section) and $e \in E_x$, Find $g\in \epsilon(X)$ and $f\in \epsilon(X,E)$ such that $dg_x = v$ and $f(x)=e$, Then we define \begin{align} \sigma_k(L) (x,v) e = L \left( \frac{i^k}{k!} (g-g(x))^k f\right) (x) \in F_x \end{align} I want to show that this defnition works, independent choice of $g$ and $f$. Can you give me any hints? The textbook says it is easy to see.... But i don't get any clue.