
theorem lemoe2:
for F being ordered Field,
    E being FieldExtension of F
for P being Ordering of F
holds QS(E,P) is prepositive_cone iff not -1.E in QS(E,P)
proof
let F be ordered Field, E be FieldExtension of F;
let P be Ordering of F;
set T = QS(E,P);
A: now assume AS: QS(E,P) is prepositive_cone;
   then reconsider E1 = E as preordered Field by REALALG1:def 16;
   reconsider O = QS(E,P) as Preordering of E1 by AS;
   not -1.E in O by REALALG1:26;
   hence not -1.E in QS(E,P);
   end;
now assume AS: not -1.E in QS(E,P);
   QS E c= T & SQ E c= QS E by REALALG1:def 13,REALALG1:18;
   then A: SQ E c= T;
   T * T c= T
     proof
     now let o be object;
       assume o in T * T;
       then consider a,b being Element of E such that
       C: o = a * b & a in T & b in T;
       thus o in T by C,REALALG1:def 5;
       end;
     hence thesis;
     end;
   then T is negative-disjoint by A,AS,REALALG1:21;
   hence QS(E,P) is prepositive_cone;
   end;
hence thesis by A;
end;
