Differentiable Manifolds/Vector (sub-)bundles, sections, foliations, distributions and Frobenius' theorem

What are vector bundles?
Lemma 10.2: Let $$E$$ be a vector bundle of the manifold $$M$$ with projection $$\pi_E$$. Then if $$p, q \in M$$ are contained in the same connected component of $$M$$, the dimensions of $$\pi_E^{-1}(p)$$ and $$\pi_E^{-1}(q)$$ are equal.

Proof:

We define the function $$\mu: M \to \mathbb N_0, \mu(q) := \text{dim } \pi_E^{-1}(q)$$. Let now $$p \in M$$ be contained in the connected component of $$M$$ $$Q$$. If we pick any point $$q \in M$$, due to the definition of a vector bundle, there is an open set $$O \subseteq M$$ with $$q \in O$$ and a bundle chart $$\psi_{E, O} : O \times \mathbb R^{k_q}$$, where $$k_q$$ is the dimension of $$$$, such that $$\psi_E(p, \cdot ) : \mathbb R^{k_q} \to E$$ has $$\pi_E^{-1}(p)$$ as its image and is a linear isomorphism. Therefore, for all $$r \in O$$, the dimension of $$\pi_E^{-1}(r)$$ is equal to $$k_q$$.

From this follows that the set $$\{q \in M | \mu(q) = \mu(p)\}$$ and its complement in $$M$$ are both open (since for every point $$q$$ in the complement of this set there also exists an open neighbourhood such that all points $$r$$ in this neighbourhood satisfy $$\text{dim } \pi_E^{-1}(r) = \text{dim } \pi_E^{-1}(q)$$ and thus all the points $$r$$ in that neighbourhood are also contained in the complement).

But if the set $$\{q \in M | \mu(q) = \mu(p)\}$$ and its complement in $$M$$ are both open, then so are the respective intersections with $$Q$$ with respect to the subspace topology on $$Q$$ induced by the topology of $$M$$. But, by definition of a connected component, $$Q$$ is a connected set with respect to the subspace topology, and thus, since $$\{q \in M | \mu(q) = \mu(p)\} \cap Q$$ is open and closed (since the complement is open) and nonempty (as it contains $$p$$), this set equals the whole set $$Q$$ by definition of connected sets.

The tangent and cotangent bundles as part of a respective vector bundle
In the following section we want to show, that both tangent and cotangent bundle with a specific atlas are manifolds, and if we define specific projections and bundle charts, we obtain vector bundles out of them and the tangent and cotangent bundles.