Let $n\in\mathbb N$, let $(M,\omega)$ be a smooth symplectic manifold of dimension $2n$, and let $F:M\to\mathbb R^n$ be a smooth map with component functions $f_i:=\pi_i\circ F\in C^\infty(M;\mathbb R)$ for $i\in\{1,\dots,n\}$, where $\pi_i:\mathbb R^n\to\mathbb R$ is the $i$th coordinate projection. For each $h\in C^\infty(M;\mathbb R)$, let $X_h\in\mathfrak X(M)$ be the Hamiltonian vector field defined by $\iota_{X_h}\omega=dh$, and define the Poisson bracket by $\{f,g\}=\omega(X_f,X_g)$. Suppose that the components of $F$ Poisson-commute, meaning $\{f_i,f_j\}=0$ for every $i,j\in\{1,\dots,n\}$. If $c\in\mathbb R^n$ is a regular value of $F$, then every [connected component](/page/Connected%20Component) of $F^{-1}(c)$ is a Lagrangian submanifold of $(M,\omega)$.