Let $(E,M,V,\pi)$ be vector bundle over $M$ with fiber $V$ and projection $\pi$. I am tryng to prove that the space of sections $\Gamma(E)$ of this vector bundle is a diffeological space. I denote the space of plots by $P$.
Let $U \subset \mathbb{R}^n$ be an open set. Given a map $\psi\colon U\times M \rightarrow E$ such that $\pi \circ \psi(u,x)=x$ where $u\in U$ and $x \in \mathbb{R}^n$, then for each $u$, $\psi_u(x)=\psi(u,x)$ is a section. So we define a plot $\chi\colon U \rightarrow \Gamma(E)$ by $\chi(u)= \phi_u(x)$.
I am following the book Diffeology for the definition of diffeological space.
Do we have as open cover of the space of sections $\Gamma(E)$ the set of plots $\chi$, that is, for every section $s \in \Gamma(E)$, we should have a plot $\chi \in P$ and $u \in U$ such that $\chi(u)=s$?
Can every section $s \in \Gamma(E)$ be written as $s(x)= \phi_u(x)$?