Corollary 2

Let E be an extension of K admitting the structure of an ordered extension of K. For an element $x \in E$ to be positive under every such structure on E, it is necessary and sufficient that x be of the form $\sum_i p_i x_i^2$, where $x_i \in E$ and the $p_i$ are positive elements of K.

The condition is obviously sufficient; it is also necessary, for (in the notation of the proof of Th. 1), if $x \notin P_0$ there exists a maximal element P of $\mathscr{M}$ such that $x \notin P$; then $-x \in P$ by the Lemma, and x is not positive under the ordering defined by P, since $x \neq 0$.