
theorem XYZb:
for F being Field,
    E being FieldExtension of F
for a being Element of E st a^2 in F
for b being Element of FAdj(F,{a})
ex c1,c2 being Element of FAdj(F,{a})
st c1 in F & c2 in F & b = c1 + @(FAdj(F,{a}),a) * c2
proof
let F be Field, E be FieldExtension of F;
let a be Element of E;
assume AS: a^2 in F;
let b be Element of FAdj(F,{a});
A2: 0.E = 0.FAdj(F,{a}) & 1.E = 1.FAdj(F,{a}) by FIELD_6:def 6;
A9: {a} is Subset of FAdj(F,{a}) & a in {a} by TARSKI:def 1,FIELD_6:35;
per cases;
suppose a in F;
  then {a} c= the carrier of F by TARSKI:def 1;
  then A1: FAdj(F,{a}) == F by FIELD_7:3;
  take b,0.FAdj(F,{a});
  thus thesis by A1;
  end;
suppose A0: not a in F;
  set K = FAdj(F,{a}), aK = @(FAdj(F,{a}),a);
  reconsider 1V = 1.E, aV = a, bV = b as Element of VecSp(K,F)
      by A9,A2,FIELD_4:def 6;
  {1.E,a} is Basis of VecSp(K,F) by A0,AS,ThBas; then
  Lin({1V,aV}) = the ModuleStr of VecSp(K,F) by VECTSP_7:def 3; then
  bV in Lin({1V,aV}); then
  consider l being Linear_Combination of {1V,aV} such that
  A1: bV = Sum l by VECTSP_7:7;
  H: F is Subfield of K & F is Subfield of E by FIELD_4:7; then
  1.E = 1.F & 1.F = 1.K by EC_PF_1:def 1; then
  A3: 1V <> aV by A0;
  {a} is Subset of FAdj(F,{a}) &a in {a} by TARSKI:def 1,FIELD_6:35; then
  J: a is FAdj(F,{a})-membered;
  the carrier of F c= the carrier of K by H,EC_PF_1:def 1; then
  reconsider b1 = l.1V, b2 = l.aV as Element of K;
  take b1,b2;
  A7: [l.1V,1V] in [:the carrier of F,the carrier of FAdj(F,{a}):]
      by A2,ZFMISC_1:def 2;
  A8: (the addF of E)||(the carrier of K) = the addF of K &
      (the multF of E)||(the carrier of K) = the multF of K by EC_PF_1:def 1;
  A4: (the lmult of VecSp(FAdj(F,{a}),F)).(l.1V,1V)
    = ((the multF of FAdj(F,{a}))|
             [:the carrier of F,the carrier of FAdj(F,{a}):]).(b1,1.E)
      by FIELD_4:def 6
   .= (the multF of FAdj(F,{a})).(b1,1.E) by A7,FUNCT_1:49
   .= ((the multF of E)||carrierFA({a})).(b1,1.E) by FIELD_6:def 6
   .= ((the multF of E)||the carrier of FAdj(F,{a})).(b1,1.E) by FIELD_6:def 6
   .= b1 * 1.FAdj(F,{a}) by A8,EC_PF_1:def 1;
  A7: [l.aV,a] in [:the carrier of F,the carrier of FAdj(F,{a}):]
      by A9,ZFMISC_1:def 2;
  A6: (the lmult of VecSp(FAdj(F,{a}),F)).(l.aV,aV)
    = ((the multF of FAdj(F,{a}))|
             [:the carrier of F,the carrier of FAdj(F,{a}):]).(b2,a)
      by FIELD_4:def 6
   .= b2 * aK by J,A7,FUNCT_1:49;
  bV = l.1V * 1V + l.aV * aV by A1,A3,VECTSP_6:18
    .= b1 + aK * b2 by A4,A6,FIELD_4:def 6;
  hence thesis;
  end;
end;
