Corollary 1

Let A, B be two commutative rings, $p : A \to B$ a homomorphism such that B is a flat A-module and E a finitely presented A-module. For every finitely generated submodule F of E, the submodule of the dual of $E_{(B)}$ orthogonal to $F_{(B)}$ is equal to $(F')_{(B)}$, where $F'$ is the submodule of the dual $E^*$ of E orthogonal to F.

By Proposition 11 $(E^*)_{(B)}$ is canonically isomorphic to the dual $(E_{(B)})^*$ of $E_{(B)}$. Then it suffices to apply Proposition 12 with $E_1 = E^*$, $E_2 = E$, $E_3 = A$, $F_2 = F$, $F_3 = \{0\}$, and $f$ the canonical bilinear form on $E^* \times E$.