
theorem lemoe1:
for F being ordered Field,
    E being FieldExtension of F
for P being Ordering of F holds P c= QS(E,P)
proof
let F be ordered Field, E be FieldExtension of F;
let P be Ordering of F;
I: F is Subfield of E by FIELD_4:7; then
H: the carrier of F c= the carrier of E by EC_PF_1:def 1;
now let o be object;
  assume AS: o in P;
  then reconsider a = o as Element of E by H;
  reconsider f = <*a*> as non empty FinSequence of E;
  A: Sum f = a by RLVECT_1:44;
  f is P-quadratic
    proof
    now let i be Element of NAT;
      assume B: i in dom f;
      dom f = {1} by FINSEQ_1:2,FINSEQ_1:38; then
      C: i = 1 by B,TARSKI:def 1;
      D: 1.F = 1.E by I,EC_PF_1:def 1;
      thus ex c being non zero Element of E,
              b being Element of E st c in P & f.i = c * b^2
        proof
        per cases;
        suppose a = 0.E;
          then 1.E * (0.E)^2 = f.i by C;
          hence thesis by D,REALALG1:25;
          end;
        suppose a <> 0.E;
          then a is non zero & a * (1.E)^2 = f.i by C;
          hence thesis by AS;
          end;
        end;
      end;
    hence thesis;
    end;
  hence o in QS(E,P) by A;
  end;
hence thesis;
end;
