Corollary 3

Suppose that $K'$ is a finitely generated extension of $K$, that $\Gamma$ is isomorphic to $\mathbf{Z}^{h'}$ (ordered lexicographically) and there is equality in formula (10). Then $\Gamma'$ is isomorphic to $\mathbf{Z}^{h'}$ (ordered lexicographically) and $k'$ is a finitely generated extension of $k$.

If there is equality in (10), there is equality in (9), whence the fact that $k'$ is a finitely generated extension of $k$ and that $\Gamma''$ is a finitely generated $\mathbf{Z}$-module. Further, comparing (9) and (10), it is seen that $h' - h = r(\Gamma'/\Gamma)$, whence $h' = r(\Gamma')$ and Proposition 4 (no. 2) then shows that $\Gamma'$ is isomorphic to $\mathbf{Z}^{h'}$ ordered lexicographically.