
theorem :: Artin
for F being formally_real Field,
    a being Element of F
holds (for O being Ordering of F holds a in O) iff a in QS F
proof
let F be formally_real Field, a be Element of F;
reconsider Q = QS F as Preordering of F;
hereby assume for O being Ordering of F holds a in O;
   then for O being Ordering of F st Q c= O holds a in O;
   then a in /\(Q,F);
   hence a in QS F by s1;
   end;
assume AS: a in QS F;
     let O be Ordering of F;
     QS F c= O by REALALG1:24;
     hence a in O by AS;
end;
