Products $s\wedge t$ generate $C_.^\infty(M,E\otimes F)$? I am reading Demially and came upon the claim that

for every $s\in C^\infty_.(M,E)$, $t\in C^\infty_.(M,F)$, the wedge product $s\wedge t$ can be combined with the bilinear map $E\times F\to E\otimes F$ in order to obtain a section $s\wedge t\in C^\infty_.(M,E\otimes F)$ of degree $\deg(s)+\deg(t)$.

How exactly is $s\wedge t\in C^\infty(M,E\otimes F)$ defined? I tried to define it in a local chart by $s_\alpha\wedge t_\alpha=\sum_{m,n,i,j} \sigma^i_{m\alpha}\theta^j_{n\alpha} s_{m\alpha}\otimes t_{n\alpha}\otimes(dz_{i\alpha}\wedge dz_{j\alpha})$ but couldn't verify that this section follows the transition relations of $C^\infty_.(M,E\otimes F)$.
Later on, the author claims that

As the product $s\wedge t$ generate $C_.^\infty(M,E\otimes F)$,

I could see why on a local chart $s\wedge t$ generate $C_.^\infty(M,E\otimes F)$, but also come to difficulties as to why is it globally true. Any help?
Also, here is a screen shot.

$D_{E \otimes F}$ will be defined in such a way that the usual formula for the differentiation of a product remains valid. For every $s \in \mathscr{C}_\bullet^\infty (M,E), t \in \mathscr{C}_\bullet^\infty(M,F)$, the wedge product $s \wedge t$ can be combined with the bilinear map $E \times F \to E \otimes F$ in order to obtain a section $s \wedge t \in \mathscr{C}^\infty(M, E \otimes F)$ of degree $\deg s + \deg t$. Then there exists a unique connection $D_{E \otimes F}$ such that $$\tag{4.2} D_{E \otimes F}(s \wedge t) = D_E s \wedge t + (-1)^{\deg s} s \wedge D_F t.$$ As the products $s \wedge t$ generate $\mathscr{C}_\bullet^\infty(M, E \otimes F)$, the uniqueness is clear. If $E, F$ are...

 A: After thinking about what Ted Shifrin hinted, I post a tentative answer for part 1 of the question. Suppose we have $s\in C^\infty_k(M,E)$, we have the corresponding multilinear (as $C^\infty(M)$ modules) alternating functions $\eta:\mathfrak{X}(M)\times ...\times\mathfrak{X}(M)\to \Gamma(M,E)$. (Morita explains the equivalence between these two forms of differential forms with value in vector bundles in page 192 of Geometry of Differential Forms.) Do the same thing for $t\in C^\infty_l(M,F)$ and we have $\mu:\mathfrak{X}(M)\times ...\times\mathfrak{X}(M)\to \Gamma(M,E)$. Now observe the natural bilinear map $\beta:\Gamma(M,E)\times\Gamma(M,F)\to\Gamma(M,E)\otimes\Gamma(M,F)\cong\Gamma(M,E\otimes F)$. We now define $s\wedge t$ by defining $\eta\wedge\mu$, which is $\eta\wedge\mu(X_1,...,X_{k+l})=\frac{1}{k!l!}\sum_{\sigma\in S_{p+q}}sgn(\sigma)\beta(\eta(X_{\sigma(1)},...,X_{\sigma(k)}),\mu(X_{\sigma(k+1)},...,X_{\sigma(k+l)}))$ for $X_i\in\mathfrak{X}(M)$. This then gives a multilinear alternating function from $\mathfrak{X}(M)\times...\mathfrak{X}(M)$ to $\Gamma(M,E\otimes F)$, which correspondingly gives a section of $C^\infty_{k+l}(M,E\otimes F)$ we call $s\wedge t$. As we go back from $\eta$ to $s$ at $p\in M$, we could define $s(x_1,...,x_k)$ with $x_i\in T_pM$ as $\eta(X_1,...,X_k)(p)$ with $X_i|_p=x_i$ by a similar reasoning as Theorem 2.8 in section 2.1 of Morita. On a chart $U_\alpha$, we therefore have $s=\sum_{I,m} \sigma_{\alpha,I,m}dz_{\alpha,I}\otimes e_{\alpha,m}$ and $t=\sum_{J,n}\beta_{\alpha,J,n}dz_{\alpha,J}\otimes f_{\alpha,n}$. By definition of $\eta\wedge\mu$, we have locally $s\wedge t=\sum_{I,J,m,n}\sigma_{\alpha,I,m}\beta_{\alpha,J,n}(e_{\alpha,m}\otimes f_{\alpha,n})\otimes dz_I\wedge dz_J$.
