  This work discusses the approximations frequently used to calculate the
contribution to pion production in proton-proton scattering near threshold from
irreducible pion re-scattering. The reference result is obtained from the
quantum mechanical reduction of the corresponding Feynman diagram to
Time-Ordered Field Theory diagrams. We conclude that the Distorted Wave Born
Approximation is quite adequate at threshold energies and even above. We
discuss the choices for the energy of the exchanged pion, both for its
propagator and for the piN amplitude describing the re-scattering vertex. The
calculation employs a physical model for nucleons and pions -- pseudo-vector
coupling for the piNN vertex and realistic amplitudes for the piN re-scattering
and for the NN transitions in the initial and final states.
