I'm trying to understand the proof of the following theorem
An integrable distribution $\cal{D}$ induces a partition of the manifold into maximal leaves.
The proof starts like this:
Take $m\in M$, define $$L_m = \{p\in M\ \vert\ \exists\text{ a path from m to p tangent to }\cal{D}\}$$
Take an atlas of foliated charts $A$ (i.e. $(U,\phi)$ is such that ${\cal{D}}_q = \left<\partial_{x^1}\vert_q,\cdots, \partial_{x^k}\vert_q \right>$) and take $(U,\phi)$ such that $L_m\cap U\ne \varnothing$. Take $p\in U$ and consider the plaque $S$ of $U$ (i.e. $S=\{q\in U\ \vert\ x^{k+1} = a^1,\cdots, x^{m} = a^{m-k}\}$ with $a^i$ constants) passing by $p$. Then $S\subset L_m$.
Why is this last fact true? I don't see any reason that $S$ should be path connected.