
theorem mainY:
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
ex g1,g2 being quadratic non empty FinSequence of F,
   g3 being non empty FinSequence of F
st Sum f =
 @(Sum g1 + b * Sum g2,FAdj(F,{a})) + @(FAdj(F,{a}),a) * @(Sum g3,FAdj(F,{a}))
proof
let F be Field, E be FieldExtension of F, 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; then
not a in F & a^2 in F; then
consider g1,g2,g3 being non empty FinSequence of FAdj(F,{a}) such that
A: g1 is quadratic non empty FinSequence of F &
   g2 is quadratic non empty FinSequence of F &
   g3 is non empty FinSequence of F &
   Sum f = Sum g1 + @(FAdj(F,{a}),a)^2 * Sum g2 + @(FAdj(F,{a}),a) * Sum g3
   by mainX;
reconsider h1 = g1, h2 = g2 as quadratic non empty FinSequence of F by A;
reconsider h3 = g3 as non empty FinSequence of F by A;
set K = FAdj(F,{a});
    {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;
take h1,h2,h3;
H: K is Subring of E by FIELD_5:12;
   F is Subfield of K by FIELD_4:7; then
I: F is Subring of K by FIELD_5:12; then
B: Sum g1 = Sum h1 & Sum g2 = Sum h2 & Sum g3 = Sum h3 by FIELD_4:2;
D: @(K,a) * @(K,a) = b by AS,H,FIELD_6:16;
F: @(K,a) * Sum g3 = @(K,a) * @(Sum h3,K) by B,FIELD_7:def 4;
   @(K,a)^2 * Sum g2 = b * Sum h2 by B,D,I,FIELD_6:16; then
Sum g1 + @(K,a)^2 * Sum g2 = Sum h1 + b * Sum h2 by B,I,FIELD_6:15;
hence thesis by F,A,FIELD_7:def 4;
end;
