Proposition 8

If $G_1, G_2$ are any closed subgroups of $\mathbf{R}^n$, we have

The real number $(u|x + y)$ is an integer for all $x \in G_1$ and all $y \in G_2$ if and only if $(u|x)$ is an integer for all $x \in G_1$ and $(u|y)$ is an integer for all $y \in G_2$, because $(u|x + y) = (u|x) + (u|y)$; hence

for any two subgroups $G_1, G_2$ of $\mathbf{R}^n$. If now we suppose that $G_1$ and $G_2$ are closed, we have $(G_1^* + G_2^*)^* = G_1 \cap G_2$ by Proposition 6; hence, taking the associated subgroups and applying Proposition 6 again,