Given a linear connection $D$ on a vector bundle $E \xrightarrow{\pi} M$ with $\mathrm{dim}M=n$ and $\mathrm{rank}E=r$, there's a distribution $\mathcal H E$ of rank $n$ over $E$ such that for every section $\sigma$ of $E$
$$ (D\sigma)_p=0 \iff \mathrm{im}\,\sigma_{*,p}=\mathcal H_{\sigma_p}$$
Moreover, if $\mathcal V E$ is the vertical subbundle of $E$ defined by $\mathcal V_\xi E=\mathrm{ker}\,\pi_{*,\xi}=T_\xi E_{\pi(\xi)}$, then one has that $TE=\mathcal V E \bigoplus \mathcal H E$.
I want to prove the following statement: $D$ is flat $\iff$ $\mathcal H E\,$ is integrable. Here's my attempt:
$(\Rightarrow)$ Let $\xi \in E$ and $p=\pi(\xi)$. I have to show an integral manifold of $\mathcal H E$ containing $\xi$. By the flatness of $D$, there is a parallel section $\sigma$ over an open neighborhood $U$ of $p$ such that $\sigma_p=\xi$. As a section, $\sigma$ is injective (1) and an immersion and so $\sigma(U)$ is an immersed submanifold and $\sigma$ is a diffeomorphism onto its image. In particular $\sigma_{*,p}$ is an isomorphism and so $T_{\sigma_p}S=\sigma_{*,p}(T_p M)=\mathcal H_{\sigma_p}E$.
$(\Leftarrow)$ Let $\xi \in E$ and $p=\pi(\xi)$ and let $S$ be an integral manifold of $\mathcal H E$ in $\xi$, so that $\mathrm dim S=n$. I have to find a parallel local section $\sigma$ near $p$ with $\sigma_p=\xi$. Consider the map $\pi|_S\colon S \to M$. Its differential $\pi_{*,\xi}$ is and isomorphism then, by the inverse function theorem, there are an open neighborhood $U$ of $p$ and an open neighborhood $V$ of $\xi$ in $S$ such that $\pi|_V \colon V \to U$ is a diffeomorphism (2). If $\sigma$ is the inverse of $\pi|_V$, then it is a local section and moreover $$\forall q \in U, \:\:\:\sigma_{*,q}(T_q M)=T_{\sigma_q}S=\mathcal H_{\sigma_q} E$$ so that $D\sigma=0$ on $U$.
Is my reasoning correct? In particular:
(1) is it true that every section is injective? (for the definition I'm using, the map defining an immersed submanifold have to be injective)
(2) Is $\pi_{*,\xi}$ an isomorphism because that $\mathrm dim M=\mathrm dim S$?