Operators of order $k$ between Sobolev spaces

Question

It may sound like tautology, but I have a problem in proving that differential operator of order $k$ is an operator of order $k$. What exactly I'm asking for? Let $M$ be a manifold and $E,F$ be two vector bundles over it (here $M$ is not assumed to be compact but in most cases it is so): then an order $k$ differential operator is an operator which can be locally expressed as $D=\sum_{|\alpha| \leq k} A_{\alpha}\frac{\partial^{\alpha}}{\partial x_1^{\alpha_1}...\partial x_n^{\alpha_n}}$ and an operator of order $k$ is understood as an operator $T:\Gamma_c^{\infty}(M,E) \to \Gamma_c^{\infty}(M,F)$ (acting on smooth, compactly supported sections) which extends to bounded operator $T:W^{s}(M,E) \to W^{s-k}(M,F)$ where $W^s$ are Sobolev spaces defined as the completion of $\Gamma^{\infty}_c$ with respect to certain norm (which is defined precisely in the discussion here. It is not to hard to verify this in local coordinates but I have a trouble to understand the global picture-in other words I don;t see why the gluing process would not spoil anything (in the noncompact case let us assume that the matrices $A_{\alpha}$ are bounded). If we compute $\|D\xi\|_{s-k,F}$ we get $\sum_i \|k_i \circ (f_i D\xi) \circ \varphi_i^{-1}\|_{s-k,\mathbb{R}^n}$. If I had not made a mistake the expression $k_i \circ (f_i D\xi) \circ \varphi_i^{-1}$ evaluated on $y=\varphi_i(x)$ is equal to $f_i(x)\sum_{|\alpha| \leq k}A_{\alpha}(\varphi_i(x))D^{\alpha}(h_i \circ \xi \circ \varphi_i^{-1})(y)$ (where $h_i$ and $k_i$ are trivialisations (to be precise, we compose it with coordinate charts) for $E$ and $F$ resp. and $f_i$ is the partition of unity). Therefore the norm of $D\xi$ is $$\|D\xi\|_{s-k},F=\sum_{i}\|(f_i \circ \varphi_i^{-1})(\sum_{|\alpha| \leq k}A_{\alpha}D^{\alpha}(h_i \circ \xi \circ \varphi_i^{-1})\|_{s-k,\mathbb{R}^n}$$ However $\|\xi\|_{s,E}=\sum_{i}\|h_i \circ \xi \circ \varphi_i^{-1}\|_{s,\mathbb{R}^n}$ and I don't see how to get an estimate $\|D\xi\|_{s-k,F} \leq C\|\xi\|_{s,E}$:
1 First of all, I see the problem where our $M$ need not to be compact: in these case the covering by charts may be infinite and the norms of "local parts" of our differential operators may not be uniformly bounded.
2 Secondly, even when our manifold $M$ is compact, how do we proceed in order to still have $f_i$'s at the end? It seems to me that if I use the local estimate, then $f_i$ is no longer on the right hand side of my inequality?
I have a problem in veryfing this and in all literature that I know authors state that this is trivial. I tried to be really precise and trace carefully in which space we are working (I mean all identifications via charts and trivialisations stuff). Therefore I will be very grateful if anyone could help me.