In this notes Geometric Wave Equations by Stefan Waldmann at page 7 he has
Let $E \longrightarrow M$ be a vector bundle of rank $N$. For a chart $(U, \psi)$ we consider a compact subset $K \subseteq U$ together with a basis sections $e_{\alpha} \in \Gamma^{\infty}\left(\left.\boldsymbol{E}\right|_{\boldsymbol{U}}\right)$ such that for a section $s$ we have
$$
\left.s\right|_{U}=s^{\alpha} e_{\alpha}
$$
We define the seminorms
$$
\mathrm{p}_{U, \psi, K, \ell,\left\{e_{\alpha}\right\}}(s)=\sup _{\substack{x \in K \\|I| \leq \ell \\ \alpha=1, \ldots, N}}\left|\frac{\partial^{|I|} s^{\alpha}}{\partial x^{I}}(x)\right|
$$
then he has
for every compact subset $K \subseteq M$ we consider another seminorm $$ \mathrm{p}_{K, \ell}(s)=\sup _{p \in K}\left\|\left.\left(\mathrm{D}^E\right)^{\ell} s\right|_p\right\|_h, $$ where $\left(\mathrm{D}^E\right)^{\ell}$ is the symmetrized covariant derivative
The goal here is to proof that $$ \mathrm{p}_{U, x, K, \ell,\left\{e_\alpha\right\}}(s) \leq c \max _{\ell^{\prime} \leq \ell} \mathrm{p}_{K, \ell^{\prime}}(s)\tag 0 $$ to do that he uses the fact that $$ \left.\left(\mathrm{D}^E\right)^{\ell} s\right|_U=\mathrm{d} x^{i_1} \vee \cdots \vee \mathrm{d} x^{i_{\ell}} \otimes e_\alpha \frac{\partial^{\ell} s^\alpha}{\partial x^{i_1} \cdots \partial x^{x_{\ell}}}+\text { (lower order terms). } \tag{**}$$
his argument at page 11 is
given a $\mathrm{p}_{U, \psi, K, \ell,\left\{e_\alpha\right\}}$ we see from $(* *)$ that we can estimate the partial derivatives $\frac{\partial^{|J|} s_s^\alpha}{\partial x^J}$ with $|J| \leq \ell$ by norm of $\left(\mathrm{D}^E\right)^{\ell} s$ and norms of partial derivatives $\frac{\partial^{\mid J^{\prime}} \mid s^\alpha}{\partial x^{J^{\prime}}}$ for $\left|J^{\prime}\right|<\ell$.
I am not seeing why his argument is true.
how can we proof eq $(0)$ for $l=1$ where have $$\left.\left(\mathrm{D}^E\right)^{\ell} s\right|_U =\left[\partial_i s^a+\left(\Gamma_E\right)_{i b}^a s^b\right] d x^i \otimes e_a \tag 1$$?