
theorem lemoe3:
for F being ordered Field,
    E being ordered FieldExtension of F
for P being Ordering of F, O being Ordering of E
st O extends P holds QS(E,P) c= O
proof
let F be ordered Field, E be ordered FieldExtension of F;
let P be Ordering of F, O be Ordering of E;
assume O extends P; then
AS: P c= O by l13;
defpred P[Nat] means
  for f being P-quadratic FinSequence of E
  st len f = $1 holds Sum f in O;
I: F is Subfield of E by FIELD_4:7;
now let f be P-quadratic FinSequence of E;
  assume len f = 0; then
  f = <*>(the carrier of E); then
  Sum f = 0.E by RLVECT_1:43 .= 0.F by I,EC_PF_1:def 1;
  hence Sum f in O by AS,REALALG1:25;
  end; then
I0: P[0];
IS: now let k be Nat;
    assume IV: P[k];
    now let f be P-quadratic FinSequence of E;
      assume B0: len f = k+1;
      then f <> {};
      then consider h being FinSequence, x being object such that
      B1: f = h^<*x*> by FINSEQ_1:46;
      reconsider f1 = f as P-quadratic non empty FinSequence of E by B0;
      per cases;
      suppose h is empty; then
      B0: f = <*x*> by B1,FINSEQ_1:34; then
      dom f = {1} by FINSEQ_1:2,FINSEQ_1:38; then
      B2: 1 in dom f by TARSKI:def 1; then
      B5: f.1 in rng f & f.1 = x by B0,FUNCT_1:3; then
      reconsider x as Element of E;
      consider a being non zero Element of E,
               b being Element of E such that
      B3: a in P & f.1 = a * b^2 by B2,dq;
      B4: Sum f = x by B0,RLVECT_1:44;
      a in O & b^2 in O by B3,AS,REALALG1:23;
      hence Sum f in O by B3,B4,B5,REALALG1:def 5;
      end;
      suppose h is non empty; then
      reconsider h as non empty FinSequence of E by B1,FINSEQ_1:36;
      reconsider r = <*x*> as non empty FinSequence of E by B1,FINSEQ_1:36;
      len r = 1 by FINSEQ_1:39;
      then B2: len f = len h + 1 by B1,FINSEQ_1:22;
      dom r = {1} by FINSEQ_1:2,FINSEQ_1:38; then
      1 in dom r by TARSKI:def 1; then
      r.1 in rng r & r.1 = x by FUNCT_1:3; then
      reconsider x as Element of E;
      f1 = h^r by B1; then
      reconsider h,r as P-quadratic FinSequence of E by XYZbS3a;
      B3: Sum r in O
          proof
          dom r = {1} by FINSEQ_1:2,FINSEQ_1:38; then
          1 in dom r by TARSKI:def 1; then
          consider a being non zero Element of E,
                   b being Element of E such that
          B3: a in P & r.1 = a * b^2 by dq;
          B4: Sum r = x by RLVECT_1:44;
          a in O & b^2 in O by B3,AS,REALALG1:23;
          hence thesis by B3,B4,REALALG1:def 5;
          end;
      Sum h in O by IV,B0,B2; then
      Sum r + Sum h in O by B3,IDEAL_1:def 1;
      hence Sum f in O by B1,RLVECT_1:41;
      end;
      end;
    hence P[k+1];
    end;
I: for k being Nat holds P[k] from NAT_1:sch 2(I0,IS);
now let o be object;
  assume o in QS(E,P); then
  consider f being P-quadratic FinSequence of E such that
  A: Sum f = o;
  consider n being Nat such that H: len f = n;
  thus o in O by H,A,I;
  end;
hence thesis;
end;
