Studying the basics of differential theory in Foundations of Mechanics 2nd edition I got confused by definitions.
A vector bundles atlas is defined as a maximum family $\{(U,\phi)\}$ of local bundle charts covering a space, and whose transition map is a linear bundle isomorphism.
There is neigther any word about existence of a homeomorphism $\Phi:\pi^{-1}(U)\to U\times \mathbb{R}^k$ nor any proof of it, meanwhile it is a condition in order to a vector bundle to be a manifold.
Absence of this constraint in definition makes me think that this can be proved, but I have no idea how. All the vector bundles manuals I found put it as a definition.
Would appreciate any help!
Upd: Lemme display the full thought in order to figure it out.
We take covering of a smooth manifold $M$ and make from it a family of local bundle charts, bijectivelly attaching to each neighborhood Cardesian product: $A=\{(u,\phi)|U\in \tau(M),\phi:U \longleftrightarrow U \times F \}$.
We put on a constraint in order to have an atlas: $\forall(U,\phi),(V,\psi)\in A \;\;\;\phi\circ\psi^{-1}\vert_{\phi(U)\cap\psi(V)}$ is a local bundle isomorphism, so by definition it's a diffeomorphism.
Then we define for vector bundles $A_1$ and $A_2$: $A_1 \sim A_2 \stackrel{def}{\iff} A_1 \cup A_2$ is vector bundle.
Then $\mathcal{A}_{max}=A/\mathord{\sim}$, and $\forall(U,\phi)\in \mathcal{A}_{max}\;\;\;U:=\pi_{\sim}(U)$.
Under certain conditions we can induce smooth structure onto $\mathcal{A}_{max}$. [https://math.stackexchange.com/questions/496571/under-what-conditions-the-quotient-space-of-a-manifold-is-a-manifold]
As far as finite-dimensional normed space $F\simeq\mathbb{R}^k, U\times F\simeq U\times \mathbb{R}^k$. The final detail to prove that $A$ is a manifold is somehow making $U$ homeomorphic to $\mathbb{R}^m$, but it is a submanifold of $M$.
How can we do this?