
theorem XYZbS3a:
for F being ordered Field,
    E being FieldExtension of F
for P being Ordering of F
for f being P-quadratic FinSequence of E
for g1,g2 being FinSequence of E
st f = g1^g2 holds g1 is P-quadratic & g2 is P-quadratic
proof
let F be ordered Field, E be FieldExtension of F, P be Ordering of F;
let f be P-quadratic FinSequence of E;
let g1,g2 be FinSequence of E;
assume AS: f = g1^g2;
now let i be Nat;
  assume F0: i in dom g1;
  then F1: g1.i = f.i by AS,FINSEQ_1:def 7;
  dom g1 c= dom f by AS,FINSEQ_1:26;
  hence ex a being non zero Element of E,
        b being Element of E st a in P & g1.i = a * b^2 by F1,F0,dq;
  end;
hence g1 is P-quadratic;
now let i be Nat;
  assume F0: i in dom g2;
  then F1: g2.i = f.(len g1 + i) by AS,FINSEQ_1:def 7;
  len g1 + i in dom f by F0,AS,FINSEQ_1:28;
  hence ex a being non zero Element of E,
        b being Element of E st a in P & g2.i = a * b^2 by F1,dq;
  end;
hence g2 is P-quadratic;
end;
