  We prove an $S_{3}$-symmetry of the Jacobi identity for intertwining operator
algebras. Since this Jacobi identity involves the braiding and fusing
isomorphisms satisfying the genus-zero Moore-Seiberg equations, our proof uses
not only the basic properties of intertwining operators, but also the
properties of braiding and fusing isomorphisms and the genus-zero Moore-Seiberg
equations. Our proof depends heavily on the theory of multivalued analytic
functions of several variables, especially the theory of analytic extensions.
