Proposition 7

The mapping $M \mapsto a(M)$ is an increasing bijection of the set of major subsets of $G$ onto the set of sub-A-modules of $K$.

Let $b$ be a sub-A-module of $K$. The set of $v(x)$ for $x \in b - (0)$ is a major subset $M(b)$ of $G$. Proposition 7 will be shown if the following equations are proved:

(2) $M(a(N)) = N$ for every major subset $N$ of $G$; (3) $a(M(b)) = b$ for every sub-A-module $b$ of $K$.

Formula (2) is easy, since, for all $m \in N$, there exists $x \in K$ such that $v(x) = m$. Then obviously $b \subset a(M(b))$; conversely, let $x \in a(M(b))$ and suppose $x \neq 0$;

then $v(x) \in M(b)$ and therefore there exists $y \in b$ such that $v(x) = v(y)$; whence $x = uy$ where $v(u) = 0$, which proves that $x \in A$ $y \subset b$ and completes the proof.