Corollary 1

Let E and E' be two subfields of N containing K; then E $\subset$ E' if and only if g(E) $\supset$ g(E'). If A and A' are two closed subgroups of $\Gamma$, then A $\subset$ A' if and only if k($\Delta$) $\supset$ k($\Delta'$).

For the two inverse bijections $E \rightleftharpoons g(E)$ and $A \rightleftharpoons k(\Delta)$ are inclusion-reversing.