In the notes of Fukaya-Oh-Ohta-Ono (Lagrangian intersection Floer theory), Chapter 10 §54.1, it is stated:

Let $L_{1}$ and $L_{2}$ be a pair of oriented Lagrangian submanifolds in $(M, \omega)$ that intersect transversely at $p_{12}$. We fix an ordering of the pair as $(L_{1} , L_{2})$. We can always choose a Darboux chart in a neighborhood $U$ of $p_{12}$, $I \colon U \to V \subset \mathbb{C}^{n}$ so that $I (p_{12}) = 0$, $$ I(L_{1} \cap U) = \mathbb{R}^{n} \cap V, \quad I(L_{2} \cap U) = \sqrt{−1} \mathbb{R}^{n} \cap V.$$ The proof follows from a version of Darboux theorem (see [Theorem 7.1, Wei71]) but strongly relies on the following well-known fact in symplectic linear algebra whose proof we omit.

Lemma 54.1. The linear symplectic group $\text{Sp} (2n)$ acts transitively on the set of transversal pairs of Lagrangian subspaces.

Theorem 7.1 from Weinstein 1971, Symplectic manifolds and their Lagrangian submanifolds, is a result about Lagrangian foliations.

Does someone have a reference for the proof of the first statement and Lemma 54.1? If not, using Theorem 7.1 from Weinstein 1971 requires a Lagrangian foliation, how it such a foliation constructed from just two transverse Lagrangians?