Proposition 7 ("Principle of Noetherian induction")

— Let E be a Noetherian set, and let F be a subset of E with the following property : if $a \in E$ is such that the relation $x > a$ implies $x \in F$, then $a \in F$. Under these conditions, F $=$ E.

Indeed, suppose E $\neq$ F; then E $-$ F has a maximal element $b$. By definition we have $x \in F$ for all $x > b$; but this implies $b \in F$, which is absurd.