Assume two above conditions. Then the product defining $i_{x,y}^{\gamma}$ converges at every point of $U$ and in fact gives an element of $Symp(U)$. Moreover, the product does not depend on the choice of path $\gamma$, and for any $x,y,z\in U\setminus W_{\cal L}$ satisfies the relation $i_{x,y}i_{y,z}=i_{x,z}$.