
theorem simpmainhelp:
for F being Field,
    K being F-finite FieldExtension of F,
    E being F-finite F-extending FieldExtension of K
for a being K-algebraic Element of E
st E == FAdj(F,{a}) holds E == FAdj(K,{a}) & K == FAdj(F,Coeff MinPoly(a,K))
proof
let F be Field, K be F-finite FieldExtension of F;
let E be F-finite F-extending FieldExtension of K;
let a be K-algebraic Element of E;
assume AS: E == FAdj(F,{a});
F: FAdj(K,{a}) = FAdj(F,{a})
   proof
   E is FieldExtension of FAdj(K,{a}) by FIELD_4:7; then
   H1: FAdj(F,{a}) is FieldExtension of FAdj(K,{a}) by AS,FIELD_13:11;
   {a} is Subset of FAdj(K,{a}) & F is Subfield of FAdj(K,{a})
      by FIELD_4:7,FIELD_6:35; then
   FAdj(K,{a}) == FAdj(F,{a}) by H1,FIELD_6:37,FIELD_4:7;
   hence thesis;
   end;
hence E == FAdj(K,{a}) by AS;
set K1 = FAdj(F,Coeff MinPoly(a,K));
K1 is Subfield of K & K is Subfield of E by FIELD_4:7; then
K1 is Subfield of E by EC_PF_1:5; then
reconsider E1 = E as F-extending FieldExtension of K1 by FIELD_4:7;
reconsider a1 = a as K1-algebraic Element of E1;
A: FAdj(F,{a1}) = FAdj(K1,{a1})
   proof
   E is FieldExtension of FAdj(K1,{a1}) by FIELD_4:7; then
   H1: FAdj(F,{a1}) is FieldExtension of FAdj(K1,{a1}) by AS,FIELD_13:11;
   {a1} is Subset of FAdj(K1,{a1}) & F is Subfield of FAdj(K1,{a1})
      by FIELD_4:7,FIELD_6:35; then
   FAdj(K1,{a1}) == FAdj(F,{a1}) by H1,FIELD_6:37,FIELD_4:7;
   hence thesis;
   end;
B: K is K1-extending & Coeff MinPoly(a,K) is Subset of K1
   by FIELD_4:7,FIELD_6:35; then
reconsider p = MinPoly(a,K) as Polynomial of K1 by FIELD_7:11;
reconsider p as Element of the carrier of Polynom-Ring K1 by POLYNOM3:def 10;
LC p = LC MinPoly(a,K) by B,FIELD_8:5
    .= 1.K by RATFUNC1:def 7
    .= 1.K1 by EC_PF_1:def 1; then
reconsider p as monic Element of the carrier of Polynom-Ring K1
    by RATFUNC1:def 7;
C: p is irreducible by B,hirr;
Ext_eval(p,a1) = Ext_eval(MinPoly(a,K),a1) by B,FIELD_7:15
              .= 0.E1 by FIELD_6:52; then
D: MinPoly(a,K) = MinPoly(a1,K1) by C,FIELD_6:52;
E: deg(E,K) = deg(FAdj(K,{a}),K) by F,AS,FIELD_7:5
           .= deg MinPoly(a,K) by FIELD_6:67
           .= deg MinPoly(a1,K1) by B,D,FIELD_4:20
           .= deg(FAdj(K1,{a1}),K1) by FIELD_6:67
           .= deg(E1,K1) by A,AS,FIELD_7:5;
reconsider K1 as Field;
reconsider K2 = K as FieldExtension of K1 by FIELD_4:7;
reconsider K2 as K1-finite FieldExtension of K1 by FIELD_7:31;
reconsider E2 = E1 as K1-extending FieldExtension of K2;
reconsider E2 as K2-finite K1-extending FieldExtension of K2 by FIELD_7:31;
deg(E2,K1) = deg(E2,K1) * deg(K2,K1) by E,FIELD_7:30;
hence thesis by FIELD_7:8,XCMPLX_1:7;
end;
