
theorem XYZc:
for F being Field,
    E being FieldExtension of F
for a being Element of E st not a in F & a^2 in F
for c1,c2,d1,d2 being Element of FAdj(F,{a})
st c1 in F & c2 in F & d1 in F & d2 in F &
   c1 + @(FAdj(F,{a}),a) * c2 = d1 + @(FAdj(F,{a}),a) * d2
holds c1 = d1 & c2 = d2
proof
let F be Field, E be FieldExtension of F, a be Element of E;
assume AS1: not a in F & a^2 in F;
let c1,c2,d1,d2 be Element of FAdj(F,{a});
assume AS2: c1 in F & c2 in F & d1 in F & d2 in F &
            c1 + @(FAdj(F,{a}),a) * c2 = d1 + @(FAdj(F,{a}),a) * d2;
set K = FAdj(F,{a}), V = VecSp(K,F), j = @(K,a);
A1: 0.K = (c1 + j * c2) - (d1 + j * d2) by AS2,RLVECT_1:15
       .= (c1 + j * c2) + (-d1 + -(j * d2)) by RLVECT_1:31
       .= (c1 + j * c2) + (-d1 + (j * (-d2))) by VECTSP_1:8
       .= j * c2 + (c1 + (-d1 + (j * (-d2)))) by RLVECT_1:def 3
       .= j * c2 + ((c1 + -d1) + (j * (-d2))) by RLVECT_1:def 3
       .= (j * c2 + (j * (-d2))) + (c1 + -d1) by RLVECT_1:def 3
       .= (c1 - d1) + j * (c2 - d2)  by VECTSP_1:def 2;
reconsider 1V = 1.K, jV = j as Element of V by FIELD_4:def 6;
reconsider c1F = c1, c2F = c2, d1F = d1, d2F = d2 as Element of F by AS2;
I0: F is Subring of K & F is Subring of E by FIELD_4:def 1; then
    -d1 = -d1F & -d2 = -d2F by FIELD_6:17; then
I1: c1 - d1 = c1F - d1F & c2 - d2 = c2F - d2F by I0,FIELD_6:15;
defpred P[object,object] means
         ($1 = 1.K & $2 = c1 - d1) or ($1 = j & $2 = c2 - d2) or
         ($1 <> 1.K & $1 <> j & $2 = 0.F);
  A: for x being object st x in the carrier of V
     ex y being object st y in the carrier of F & P[x,y]
     proof
     let x be object;
     assume x in the carrier of V;
     per cases;
     suppose x = 1.K;
        hence ex y being object st y in the carrier of F & P[x,y] by I1;
        end;
     suppose x = j;
       hence ex y being object st y in the carrier of F & P[x,y] by I1;
       end;
     suppose x <> 1.K & x <> j;
       hence ex y being object st y in the carrier of F & P[x,y];
       end;
     end;
consider l being Function of the carrier of V,the carrier of F such that
L: for x being object st x in the carrier of V holds P[x,l.x]
   from FUNCT_2:sch 1(A);
reconsider l as Element of Funcs(the carrier of V, the carrier of F)
   by FUNCT_2:8;
for v being Element of V st not v in {1V,jV} holds l.v = 0.F
      proof
      let v being Element of V;
      assume not v in {1V,jV};
      then v <> 1.K & v <> j by TARSKI:def 2;
      hence l.v = 0.F by L;
      end;
then reconsider l as Linear_Combination of V by VECTSP_6:def 1;
now let o be object;
  assume o in Carrier l;
  then consider v being Element of V such that
  A1: o = v & l.v <> 0.F by VECTSP_6:1;
  (v = 1.K & l.v = c1 - d1) or (v = j & l.v = c2 - d2) by L,A1;
  hence o in {1V,jV} by A1,TARSKI:def 2;
  end;
then Carrier(l) c= {1V,jV};
then reconsider l as Linear_Combination of {1V,jV} by VECTSP_6:def 4;
a in {a} & {a} is Subset of K by TARSKI:def 1,FIELD_6:35; then
I8: a is K-membered;
I9: 1.K = 1.F by I0,C0SP1:def 3 .= 1.E by I0,C0SP1:def 3;
    1.K = 1.F by I0,C0SP1:def 3; then
I3: 1.K <> j by I8,AS1;
I2: {1V,jV} is linearly-independent by I8,I9,AS1,ThBas;
    I8: [c1-d1,1.K] in [:the carrier of F,the carrier of K:]
        by I1,ZFMISC_1:def 2;
I6: l.1V * 1V
       = ((the multF of K)|[:the carrier of F,the carrier of K:]).(l.1V,1.K)
         by FIELD_4:def 6
      .= ((the multF of K)|[:the carrier of F,the carrier of K:]).(c1-d1,1.K)
         by I3,L
      .= (c1 - d1) * 1.K by I8,FUNCT_1:49;
    I8: [c2-d2,j] in [:the carrier of F,the carrier of K:]
        by I1,ZFMISC_1:def 2;
I7: l.jV * jV
       = ((the multF of K)|[:the carrier of F,the carrier of K:]).(l.jV,j)
         by FIELD_4:def 6
      .= ((the multF of K)|[:the carrier of F,the carrier of K:]).(c2-d2,j)
         by I3,L
      .= (c2 - d2) * j by I8,FUNCT_1:49
      .= j * (c2 - d2);
    l.1V * 1V + l.jV * jV
        = (the addF of K).(l.1V*1V,l.jV*jV) by FIELD_4:def 6
       .= 0.V by A1,I6,I7,FIELD_4:def 6; then
I5: l.1V = 0.F & l.jV = 0.F by I2,I3,VECTSP_7:6;
    l.1V = c1 - d1 by I3,L; then
    c1 - d1 = 0.K by I5,I0,C0SP1:def 3;
hence c1 = d1 by RLVECT_1:21;
    l.jV = c2 - d2 by I3,L; then
    c2 - d2 = 0.K by I5,I0,C0SP1:def 3;
hence c2 = d2 by RLVECT_1:21;
end;
