Proof. The graph of $L=dK$ is a Lagrangian submanifold in $T^{\ast}{{\bf R}}^{n}={{\bf R}}^{n}\oplus({{\bf R}}^{n})^{\ast}$. Let $p_{1}$ and $p_{2}$ be the natural projections to the direct summands. They are local diffeomorphisms. Since $K$ is defined up to the adding of an affine function, the graph itself is defined up to translations. The Monge-Ampère equation corresponds to the condition $p_{1}^{\ast}(vol_{{\bf R}^{n}})=p_{2}^{\ast}(vol_{{\bf R}^{\ast n}})$, where $vol_{{\bf R}^{n}}$ (resp. $vol_{{\bf R}^{\ast n}}$) denotes the standard volume form on ${\bf R}^{n}$ (resp. ${\bf R}^{\ast n}$). The manifold $L$ can be considered as a graph of $d\widehat{K}$. Thus $\widehat{K}$ satisfies the Monge-Ampère equation as well. The Lemma is proved. $\blacksquare$