Formally Real Fields

We extend the algebraic theory of ordered fields [7, 6] in Mizar [1, 2, 3]: we show that every preordering can be extended into an ordering, i.e. that formally real and ordered fields coincide.We further prove some characterizations of formally real fields, in particular the one by Artin and Schreier using sums of squares [4]. In the second part of the article we define absolute values and the square root function [5].