reserve
F for non 2-characteristic non quadratic_complete polynomial_disjoint Field;

theorem
for a being non square Element of F
for a1,a2,b1,b2 being F-membered Element of FAdj(F,{sqrt a})
st a1 + @(sqrt a) * b1 = a2 + @(sqrt a) * b2 holds a1 = a2 & b1 = b2
proof
let a be non square Element of F;
let a1,a2,b1,b2 be F-membered Element of FAdj(F,{sqrt a});
set E = FAdj(F,{sqrt a}), j = @(sqrt a);
set V = VecSp(E,F);
assume a1 + @(sqrt a) * b1 = a2 + @(sqrt a) * b2; then
AS: 0.E = (a1 + j * b1) - (a2 + j * b2) by RLVECT_1:15
       .= (a1 + j * b1) + (-a2 + -(j * b2)) by RLVECT_1:31
       .= (a1 + j * b1) + (-a2 + (j * (-b2))) by VECTSP_1:8
       .= j * b1 + (a1 + (-a2 + (j * (-b2)))) by RLVECT_1:def 3
       .= j * b1 + ((a1 + -a2) + (j * (-b2))) by RLVECT_1:def 3
       .= (j * b1 + (j * (-b2))) + (a1 + -a2) by RLVECT_1:def 3
       .= (a1 - a2) + j * (b1 - b2) by VECTSP_1:def 2;
reconsider 1V = 1.E, jV = j as Element of V by FIELD_4:def 6;
reconsider a1R = a1, a2R = a2, b1R = b1, b2R = b2 as Element of F
    by FIELD_7:def 5;
I0: F is Subring of E by FIELD_4:def 1; then
    -a2 = -a2R & -b2 = -b2R by FIELD_6:17; then
I1: a1 - a2 = a1R - a2R & b1 - b2 = b1R - b2R by I0,FIELD_6:15;
defpred P[object,object] means
         ($1 = 1.E & $2 = a1 - a2) or ($1 = j & $2 = b1 - b2) or
         ($1 <> 1.E & $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.E;
        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.E & 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.E & 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.E & l.v = a1 - a2) or (v = j & l.v = b1 - b2) 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;
    1.E = 1.F by I0,C0SP1:def 3; then
I3: 1V <> jV by FIELD_7:def 5;
    {1.F, sqrt a} = {1V,jV} by I0,C0SP1:def 3; then
I2: {1V,jV} is linearly-independent by qbase4;
    I8: [a1-a2,1.E] in [:the carrier of F,the carrier of E:]
        by I1,ZFMISC_1:def 2;
I6: l.1V * 1V
       = ((the multF of E)|[:the carrier of F,the carrier of E:]).(l.1V,1.E)
         by FIELD_4:def 6
      .= ((the multF of E)|[:the carrier of F,the carrier of E:]).(a1-a2,1.E)
         by I3,L
      .= (a1 - a2) * 1.E by I8,FUNCT_1:49;
    I8: [b1-b2,j] in [:the carrier of F,the carrier of E:]
        by I1,ZFMISC_1:def 2;
I7: l.jV * jV
       = ((the multF of E)|[:the carrier of F,the carrier of E:]).(l.jV,j)
         by FIELD_4:def 6
      .= ((the multF of E)|[:the carrier of F,the carrier of E:]).(b1-b2,j)
         by I3,L
      .= (b1 - b2) * j by I8,FUNCT_1:49
      .= j * (b1 - b2) by GROUP_1:def 12;
    l.1V * 1V + l.jV * jV
        = (the addF of E).(a1-a2,j*(b1-b2)) by I6,I7,FIELD_4:def 6
       .= 0.V by AS,FIELD_4:def 6; then
I5: l.1V = 0.F & l.jV = 0.F by I2,I3,VECTSP_7:6;
    l.1V = a1 - a2 by I3,L; then
    a1 - a2 = 0.E by I5,I0,C0SP1:def 3;
hence a1 = a2 by RLVECT_1:21;
    l.jV = b1 - b2 by I3,L; then
    b1 - b2 = 0.E by I5,I0,C0SP1:def 3;
hence b1 = b2 by RLVECT_1:21;
end;
