  We prove a generalized rationality property and a new identity that we call
the ``Jacobi identity'' for intertwining operator algebras. Most of the main
properties of genus-zero conformal field theories, including the main
properties of vertex operator algebras, modules, intertwining operators,
Verlinde algebras, and fusing and braiding matrices, are incorporated into this
identity. Together with associativity and commutativity for intertwining
operators proved by the author in \cite{H2} and \cite{H3}, the results of the
present paper solve completely the problem of finding a natural purely
algebraic structure on the direct sum of all inequivalent irreducible modules
for a suitable vertex operator algebra. Two equivalent definitions of
intertwining operator algebra in terms of this Jacobi identity are given.
