Corollary 2

— If $a$ is a positive element of an ordered field $K$, then every splitting field $E$ of the polynomial $X^2 - a$ admits the structure of an ordered extension of $K$.

The result is trivial if $a$ is a square in $K$. Otherwise the polynomial $f(X) = X^2 - a$ is irreducible and changes sign, since $f(0) < 0$ and $f(x)$ has the same sign as $x^2$, so positive, for $x$ in the complement of some interval of $K$. We can now complete the proof by applying Prop. 3.