
theorem lemoe4:
for F being ordered Field,
    E being FieldExtension of F
for P being Ordering of F
holds P extends_to E iff QS(E,P) is prepositive_cone
proof
let F be ordered Field, E be FieldExtension of F, P be Ordering of F;
A: now assume P extends_to E; then
   consider O being Subset of E such that
   B0: P c= O & O is positive_cone;
   reconsider E1 = E as ordered FieldExtension of F by B0,REALALG1:def 17;
   reconsider O as Ordering of E1 by B0;
   B1: QS(E,P) c= O by B0,l13,lemoe3;
   not -1.E in O by REALALG1:26;
   hence QS(E,P) is prepositive_cone by B1,lemoe2;
   end;
now assume B0: QS(E,P) is prepositive_cone; then
   reconsider E1 = E as preordered Field by REALALG1:def 16;
   reconsider Pr = QS(E,P) as Preordering of E1 by B0;
   consider O being Ordering of E1 such that
   B1: Pr c= O by REALALG2:31;
   P c= Pr by lemoe1; then
   P c= O by B1;
   hence P extends_to E;
   end;
hence thesis by A;
end;
