
theorem maina:
for F being Field,
    E being FieldExtension of F
for a being Element of E, b being Element of F
for f being quadratic non empty FinSequence of FAdj(F,{a})
st not a in F & a^2 = b & Sum f in F
ex g1,g2 being quadratic non empty FinSequence of F
st Sum f = Sum g1 + b * Sum g2
proof
let F be Field, E be FieldExtension of F;
let a be Element of E, b be Element of F;
let f be quadratic non empty FinSequence of FAdj(F,{a});
assume AS: not a in F & a^2 = b & Sum f in F; then
AS1: not a in F & a^2 in F;
set K = FAdj(F,{a});
H0: F is Subring of K & K is Subring of E & F is Subring of E
    by FIELD_4:def 1,FIELD_5:12;
    {a} is Subset of K & a in {a} by TARSKI:def 1,FIELD_6:35; then
    a is K-membered; then
reconsider a as K-membered Element of E;
reconsider Sf = Sum f as Element of F by AS;
consider g1,g2 being quadratic non empty FinSequence of F,
         g3 being non empty FinSequence of F such that
A: Sum f = @(Sum g1 + b * Sum g2,K) + @(K,a) * @(Sum g3,K) by AS,mainY;
B: Sum f = @(Sf,K) + @(K,a) * 0.K by FIELD_7:def 4;
@(Sum g1 + b * Sum g2,K) = Sum f
   proof
   D: @(Sf,K) = Sf & 0.K = 0.F & Sum g1 + b * Sum g2 = @(Sum g1 + b * Sum g2,K)
      & @(Sum g3,K) = Sum g3 by H0,C0SP1:def 3,FIELD_7:def 4; then
   @(Sum g1 + b * Sum g2,K) in F & Sum g3 in F & @(Sf,K) in F & 0.K in F;
   hence thesis by B,D,AS1,A,XYZc;
   end;
hence thesis by FIELD_7:def 4;
end;
