I want to show the following:
Let $L_0,L_1 \subset \mathbb{R}^{2n}$ be Lagrangian submanifolds (standard symplectic structure on $\mathbb{R}^{2n}$. Let $p=(x,y) \in \mathbb{R}^{2n}$, s.t. $p \in L_0 \cap L_1$ and $T_pL_0 \cap T_p L_1= \{0\}$. Then there exists a symplectomorphism
$\varphi: \mathbb{R}^{2n} \rightarrow \mathbb{R}^{2n}$
such that
$\varphi(L_0 \cup L_1)\cap B_{\epsilon}(p)= (\mathbb{R}^{n} \times\{y\} \cup \{x\}\times \mathbb{R}^{n}) \cap B_{\epsilon}(p)$ for sufficient small $\epsilon$.
My idea is the following:
There are neighborhoods $U_0$ of $p$ in $L_0$ and $U_1$ of $p$ in $L_1$, and charts $\phi:U_0 \rightarrow B_{\epsilon}(x) \times \{y\}$ for $L_0$ and $\psi:U_1 \rightarrow \{x\} \times B_{\epsilon}(y)$ for $L_1$ (also $U_0 \cap U_1=\{p\}$). (Such neighborhoods exist, because $T_pL_0 \cap T_p L_1= \{0\}$)
W.l.o.g. $\phi(p)=p, \psi(p)=p$. Then set
$\varphi(q)=\begin{cases} \phi(q) & q \in U_0 \\ \psi(q)& q \in U_1 \end{cases}$
Now I'm not quite sure about my statements here and also I don't show how to extend this $\varphi$ and also how to show that it is a symplectomorphism.
Does somebody have any hints for me?