Where is the uniqueness of smooth structure for involutive distributions proved?

[center o bass]

I'm looking to prove the Global Frobenius theorem, however in order to do so I need to prove the following lemma:

If $D$ is an involutive distribution and and $\left\{N_\alpha\right\}$ is collection of integral manifolds of $D$ with a point in common, then $N = \cup_\alpha N_{\alpha}$ has a unique smooth structure making it into connected integral manifold of $D$ in which each $N_\alpha$ is an open submanifold.

Do you know somewhere where it is proved? Or can you help me prove it?

 

[Greg Bernhardt]

I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?