In Gresho and Sani (Int. J. Numer. Methods Fluids 1987; 7:1111-1145;
Incompressible Flow and the Finite Element Method, vol. 2. Wiley: New
York, 2000) was proposed an important hypothesis regarding the
pressure Poisson equation (PPE) for incompressible flow: Stated there
but not proven was a so-called equivalence theorem (assertion) that
stated/asserted that if the Navier-Stokes momentum equation is solved
simultaneously with the PPE whose boundary condition (BC) is the
Neumann condition obtained by applying the normal component of the
momentum equation on the boundary on which the normal component of
velocity is specified as a Dirichlet BC, the solution (u, p) would be
exactly the same as if the primitive equations, in which the PPE plus
Neumann BC is replaced by the usual divergence-free constraint,
were solved instead.  This issue is explored in sufficient detail
in this paper so as to actually prove the theorem for at least some
situations. Additionally, like the original/primitive equations that
require no BC for the pressure, the new results establish the same
thing when the PPE approach is employed.