  We establish a product formula for Gromov-Witten invariants for closed,
connected, relatively semi-positive Hamiltonian fibrations over any symplectic
base. Furthermore, we show that the fibration projection induces a locally
trivial (orbi-)fibration map from the moduli space of pseudo-holomorphic maps
with marked points in the total space of the Hamiltonian fibration to the
corresponding moduli space of pseudo-holomorphic maps with marked points in the
base. We use this induced map to recover the product formula by means of
integration. Finally, we give applications to c-splitting and symplectic
uniruledness.
