How to show that a manifold is foliated by stable manifolds?

Question

I'm trying to understand how to show explicitly that a manifold may be foliated by the stable manifolds, using the following statements of stable manifold theorem and foliations. I feel like it should be obvious, but I'm not confident enough in my differential geometry to know that I'm right.

STABLE MANIFOLD THEOREM:

Suppose flow $\phi_t:\mathbb{R}\times M \rightarrow M$ with a hyperbolic invariant set $\Lambda$. Let us define:

• $W^{s}(x)= \{y \in M : \displaystyle{\lim_{t\rightarrow\infty}} \|\phi_{t}(y) - \phi_{t}(x)\| =0\}$
• $W^{u}(x)= \{y \in M : \displaystyle{\lim_{t\rightarrow - \infty}} \|\phi_{t}(y) - \phi_{t}(x)\| =0\}$

Then,

1. $W^s(x)$ and $W^u(x)$ are smooth manifolds, which are called the stable manifold and unstable manifold respectively.
2. $W^s(x)$ and $W^u(x)$ are smooth injective immersions of the bundles $E^s_x$ and $E^u_x$ respectively
3. $W^s(x)$ and $W^u(x)$ are tangent to the bundles at $x$: $T(W^s(x))_x =E^s_x$ and $T(W^u(x))_x =E^u_x$.

FOLIATIONS (adapted from this article by Lawson https://projecteuclid.org/euclid.bams/1183535509):

An $p$-dimensional, $C^{k}$ foliation of an $n$-dimensional manifold $M$ is a decomposition of $M$ into a union of disjoint connected subsets $ \{\mathcal{L}_{\alpha} \}_{\alpha \in A}$ called the leaves of the foliation, with the following property: Every point in $M$ has a neighbourhood $U$ and a system of local $C^{k}$ co-ordinates $x=(x_1, ..., x_n):U\rightarrow \mathbb{R}^n$ such that for each leaf $\mathcal{L}_{\alpha}$, the components of $U\cap \mathcal{L}_{\alpha}$ are described by the equations ($x_{p+1}$, ..., $x_n$) = ($\alpha_{p+1}$, ..., $\alpha_{n}$), where $\alpha_{p+1},..., \alpha_{n}$ are constant values in $\mathbb{R}$.

I know how to show that the $M$ decomposes into a disjoint union of stable manifolds, but I don't know how to define the charts explicitly as per the definition of foliations. (If you know of better definitions, do let me know!) I was thinking of trying to construct submersion of $M$ into $W^s(x)$, because submersions (according to Lawson's article) gives us a foliation by using the Implicit Function Theorem. Not too sure how to go about this though... this is a fact used a lot in the literature but I can't seem to find a simple way of applying the Stable Manifold Theorem to show this is indeed the case.

Any help would be much appreciated!