
theorem
FAdj(F_Rat,{zeta}) = FAdj(F_Rat,{sqrt(-3)})
proof
set F = FAdj(F_Rat,{sqrt(-3)});
A: F_Rat is Subfield of FAdj(F_Rat,{sqrt(-3)}) by FIELD_6:36;
{zeta} is Subset of FAdj(F_Rat,{sqrt(-3)})
   proof
   now let o be object;
     assume o in {zeta}; then
     B1: o = zeta by TARSKI:def 1;
     B2: {sqrt(-3)} is Subset of FAdj(F_Rat,{sqrt(-3)}) by FIELD_6:35;
     sqrt(-3) in {sqrt(-3)} by TARSKI:def 1; then
     reconsider sqt = sqrt(-3) as Element of FAdj(F_Rat,{sqrt(-3)}) by B2;
     B3: F is Subring of F_Complex by FIELD_5:12; then
     B4: 1.F_Complex = 1.F & 1 = 1.F_Complex
         by COMPLEX1:def 4,COMPLFLD:def 1,C0SP1:def 3; then
     B5: -1.F_Complex = -1.F & -1 = -1.F_Complex by B3,FIELD_6:17,COMPLFLD:2;
     B6: (-1) + <i> * (sqrt 3) = (-1.F_Complex) + sqrt(-3) &
         (-1.F_Complex) + sqrt(-3) = (-1.F) + sqt by B3,B5,FIELD_6:15;
     set t = 1.F_Complex + 1.F_Complex;
     B9: t" = 2" by B4,COMPLFLD:5,COMPLFLD:9;
     B10: t = 1.F + 1.F by B4,B3,FIELD_6:15;
          t is non zero by COMPLFLD:9; then
     B11: t" = (1.F + 1.F)" by B10,FIELD_6:18;
     ((-1) + <i> * (sqrt 3)) * 2"
           = ((-1.F_Complex) + sqrt(-3)) * t" by B9,B4,COMPLFLD:2
          .= ((-1.F) + sqt) * (1.F + 1.F)" by B3,B6,B11,FIELD_6:16;
     hence o in the carrier of FAdj(F_Rat,{sqrt(-3)}) by B1;
     end;
   hence thesis by TARSKI:def 3;
   end; then
C: FAdj(F_Rat,{zeta}) is Subfield of FAdj(F_Rat,{sqrt(-3)}) by A,FIELD_6:37;
set F = FAdj(F_Rat,{zeta});
A: F_Rat is Subfield of FAdj(F_Rat,{zeta}) by FIELD_6:36;
{sqrt(-3)} is Subset of FAdj(F_Rat,{zeta})
   proof
   now let o be object;
     assume o in {sqrt(-3)}; then
     B1: o = sqrt(-3) by TARSKI:def 1;
     B2: {zeta} is Subset of FAdj(F_Rat,{zeta}) by FIELD_6:35;
     zeta in {zeta} by TARSKI:def 1; then
     reconsider zet = zeta as Element of FAdj(F_Rat,{zeta}) by B2;
     B3: F is Subring of F_Complex by FIELD_5:12; then
     B4: 1.F_Complex = 1.F & 1 = 1.F_Complex
         by COMPLEX1:def 4,COMPLFLD:def 1,C0SP1:def 3;
     set t = 1.F_Complex + 1.F_Complex;
     t = 1.F + 1.F by B3,B4,FIELD_6:15; then
     t * zeta = (1.F + 1.F) * zet by B3,FIELD_6:16; then
     t * zeta + 1.F_Complex = (1.F + 1.F) * zet + 1.F by B3,B4,FIELD_6:15;
     hence o in the carrier of FAdj(F_Rat,{zeta}) by B1,B4;
     end;
   hence thesis by TARSKI:def 3;
   end; then
FAdj(F_Rat,{sqrt(-3)}) is Subfield of FAdj(F_Rat,{zeta}) by A,FIELD_6:37;
hence thesis by C,FIELD_7:def 2,FIELD_7:2;
end;
