Is the following statement well known?

Let $M,N$ be symplectic (algebraic) manifolds. Let $L \subset M \times N$ be a (smooth) Lagrangian correspondence. For a subset $X \subset M$ we denote $L(X):=\{y\in N|(x,y)\in L\}$. Let $X \subset M$ be an isotropic subvariety (i.e. its smooth locus is isotropic). Then $L(X)$ is isotropic.

Is it written somewhere? I think I can prove it and it is quite simple, but I'll rather use a reference instead.

Thank you

Is the following statement well known?

Let $M,N$ be symplectic (algebraic) manifolds. Let $L \subset M \times N$ be a (smooth) Lagrangian correspondence. For a subset $X \subset M$ we denote $L(X):=(y\in N|(x,y)\in L)$. Let $X \subset M$ be an isotropic subvariety (i.e. its smooth locus is isotropic). Then $L(X)$ is isotropic.

Is it written somewhere? I think I can prove it and it is quite simple, but I'll rather use a reference instead.

Thank you