How to show that $\theta=\sum_{j=1}^n f_j\cdot (\alpha_j\circ \Phi)$ for some $f_j\in C^\infty(B^\prime)$ and $\alpha_j\in\Gamma(E)$?

Question

Let $\pi:E\longrightarrow B$ be a vector bundle, $\Phi:B^\prime\longrightarrow B$ and $\theta:B^\prime\longrightarrow E$ be smooth maps such that $$\pi\circ \theta=\Phi.$$ How can I show $\theta$ has the form $$\theta=\sum_{j=1}^n f_j\cdot (\alpha_j\circ \Phi),$$ for some $f_j\in C^\infty(B^\prime)$ and $\alpha_j\in \Gamma(E)$? You can add hypothesis in order to use partition of unity if you wish.

Thanks.