
theorem
for F being ordered Field,
    E being FieldExtension of F
for P being Ordering of F
holds P extends_to E iff
      for f being P-quadratic non empty FinSequence of E
      st Sum f = 0.E holds f is trivial
proof
let F be ordered Field, E be FieldExtension of F, P be Ordering of F;
Z: 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;
   now let f be P-quadratic non empty FinSequence of E;
     assume C0: Sum f = 0.E & f is non trivial;
     then Sum f in O \ {0.E} by B0,l13,lemB3;
     then Sum f in O & not Sum f in {0.E} by XBOOLE_0:def 5;
     hence contradiction by C0,TARSKI:def 1;
     end;
   hence for f being P-quadratic non empty FinSequence of E
         st Sum f = 0.E holds f is trivial;
   end;
set g = <*1.E*>;
F is Subfield of E by FIELD_4:7; then
A: 1.E = 1.F by EC_PF_1:def 1;
B: dom g = { 1 } by FINSEQ_1:2,FINSEQ_1:38;
now let i be Element of NAT;
  assume i in dom g; then
  C: i = 1 by B,TARSKI:def 1;
  1.E * (1.E)^2 = g.i by C;
  hence ex a being non zero Element of E,
           b being Element of E st a in P & g.i = a * b^2 by A,REALALG1:25;
  end; then
reconsider g as P-quadratic non empty FinSequence of E by dq;
now assume AS: for f being P-quadratic non empty FinSequence of E
   st Sum f = 0.E holds f is trivial;
   now assume -1.E in QS(E,P); then
     consider h being P-quadratic FinSequence of E such that
     C: Sum h = -1.E;
     Sum(g^h) = Sum g + Sum h by RLVECT_1:41
             .= 1.E + -1.E by C,RLVECT_1:44
             .= 0.E by RLVECT_1:5;
     hence contradiction by AS;
     end;
   hence P extends_to E by lemoe2,lemoe4;
   end;
hence thesis by Z;
end;
