
theorem
{ 1, 3-Root(2), 3-Root(2)^2, zeta, 3-Root(2) * zeta, 3-Root(2)^2 * zeta }
is Basis of VecSp(FAdj(F_Rat,{3-CRoot(2),zeta}),F_Rat)
proof
set F = FAdj(F_Rat,{3-CRoot(2), zeta}), K = FAdj(F_Rat,{3-CRoot(2)});
B: K = FAdj(F_Rat,{3-Root(2)}) by mmk;
C: FAdj(F_Rat,{3-CRoot(2), zeta})
     = FAdj(F_Rat,{3-CRoot(2)}\/{zeta}) by ENUMSET1:1
    .= FAdj(K,{zeta}) by FIELD_7:35;
set M = {1,3-Root(2),3-Root(2)^2,zeta,3-Root(2)*zeta,3-Root(2)^2*zeta};
reconsider B1 = {1,3-Root(2),3-Root(2)^2} as Basis of VecSp(K,F_Rat)
   by B,bas3R;
reconsider B2 = {1,zeta} as Basis of VecSp(F,K) by baszeta;
E: Base(B1,B2) = {a*b where a,b is Element of F : a in B1 & b in B2}
     by FIELD_7:def 7;
   Base(B1,B2) = M
     proof
     E1: now let o be object;
         assume o in M; then
         per cases by ENUMSET1:def 4;
         suppose E0: o = 1;
           set a = 1.F, b = 1.F;
           E1: a = 1.F_Complex & b = 1.F_Complex by EC_PF_1:def 1; then
               a = 1 & b = 1 by COMPLEX1:def 4,COMPLFLD:def 1; then
           E2: a in B1 & b in B2 by ENUMSET1:def 1,TARSKI:def 2;
           a * b = 1 by E1,COMPLEX1:def 4,COMPLFLD:def 1;
           hence o in Base(B1,B2) by E,E2,E0;
           end;
         suppose E0: o = 3-Root(2);
             {3-CRoot(2), zeta} is Subset of F &
             3-Root(2) in {3-CRoot(2), zeta} by TARSKI:def 2,FIELD_6:35; then
           reconsider a = 3-Root(2) as Element of F;
           set b = 1.F;
               b = 1.F_Complex by EC_PF_1:def 1
                .= 1 by COMPLEX1:def 4,COMPLFLD:def 1; then
           E1: a in B1 & b in B2 by ENUMSET1:def 1,TARSKI:def 2;
           a * b = 3-Root(2);
           hence o in Base(B1,B2) by E,E1,E0;
           end;
         suppose E0: o = 3-Root(2)^2;
             {3-CRoot(2), zeta} is Subset of F &
             3-CRoot(2) in {3-CRoot(2), zeta} by TARSKI:def 2,FIELD_6:35; then
             reconsider c = 3-CRoot(2) as Element of F;
                F is Subring of F_Complex by FIELD_5:12; then
             A: 3-CRoot(2) * 3-CRoot(2) = c * c by FIELD_6:16;
             3-Root(2)^2 = 3-Root(2) * 3-Root(2) by O_RING_1:def 1
                        .= 3-CRoot(2) * 3-CRoot(2); then
           reconsider a = 3-Root(2)^2 as Element of F by A;
           set b = 1.F;
               b = 1.F_Complex by EC_PF_1:def 1
                .= 1 by COMPLEX1:def 4,COMPLFLD:def 1; then
           E1: a in B1 & b in B2 by ENUMSET1:def 1,TARSKI:def 2;
           a * b = 3-Root(2)^2;
           hence o in Base(B1,B2) by E,E1,E0;
           end;
         suppose E0: o = zeta;
             {3-CRoot(2), zeta} is Subset of F &
              zeta in {3-CRoot(2), zeta} by TARSKI:def 2,FIELD_6:35; then
           reconsider b = zeta as Element of F;
           set a = 1.F;
               a = 1.F_Complex by EC_PF_1:def 1
                .= 1 by COMPLEX1:def 4,COMPLFLD:def 1; then
           E1: a in B1 & b in B2 by ENUMSET1:def 1,TARSKI:def 2;
           a * b = zeta;
           hence o in Base(B1,B2) by E,E1,E0;
           end;
         suppose E0: o = 3-Root(2)*zeta;
             {3-CRoot(2), zeta} is Subset of F &
              zeta in {3-CRoot(2), zeta} &
              3-Root(2) in {3-CRoot(2), zeta} by TARSKI:def 2,FIELD_6:35; then
           reconsider a = 3-Root(2), b = zeta as Element of F;
           E1: a in B1 & b in B2 by ENUMSET1:def 1,TARSKI:def 2;
           F is Subring of F_Complex by FIELD_5:12; then
           a * b = 3-CRoot(2) * zeta by FIELD_6:16;
           hence o in Base(B1,B2) by E,E1,E0;
           end;
         suppose E0: o = 3-Root(2)^2*zeta;
             A: {3-CRoot(2), zeta} is Subset of F &
              zeta in {3-CRoot(2), zeta} &
              3-Root(2) in {3-CRoot(2), zeta} by TARSKI:def 2,FIELD_6:35; then
             reconsider c = 3-CRoot(2) as Element of F;
                F is Subring of F_Complex by FIELD_5:12; then
             B: 3-CRoot(2) * 3-CRoot(2) = c * c by FIELD_6:16;
             C: 3-Root(2)^2 = 3-Root(2) * 3-Root(2) by O_RING_1:def 1
                           .= 3-CRoot(2) * 3-CRoot(2); then
           reconsider a = 3-Root(2)^2, b = zeta as Element of F by A,B;
           E1: a in B1 & b in B2 by ENUMSET1:def 1,TARSKI:def 2;
           F is Subring of F_Complex by FIELD_5:12; then
           a * b = (3-CRoot(2) * 3-CRoot(2)) * zeta by C,FIELD_6:16;
           hence o in Base(B1,B2) by C,E,E1,E0;
           end;
         end;
     now let o be object;
       assume o in Base(B1,B2); then
       consider a,b being Element of F such that
       E0: o = a * b & a in B1 & b in B2 by E;
       per cases by E0,ENUMSET1:def 1;
       suppose E1: a = 1;
         per cases by E0,TARSKI:def 2;
         suppose b = 1; then
           E2: a = 1.F_Complex & b = 1.F_Complex
                           by E1,COMPLEX1:def 4,COMPLFLD:def 1;
           F is Subring of F_Complex by FIELD_5:12; then
           a * b = 1.F_Complex * 1.F_Complex by E2,FIELD_6:16
                .= 1 by COMPLEX1:def 4,COMPLFLD:def 1;
           hence o in M by E0,ENUMSET1:def 4;
           end;
         suppose E2: b = zeta;
           E3: a = 1.F_Complex by E1,COMPLEX1:def 4,COMPLFLD:def 1;
           F is Subring of F_Complex by FIELD_5:12; then
           a * b = 1.F_Complex * zeta by E2,E3,FIELD_6:16 .= zeta;
           hence o in M by E0,ENUMSET1:def 4;
           end;
         end;
       suppose E1: a = 3-Root(2);
         per cases by E0,TARSKI:def 2;
         suppose b = 1; then
           E2: b = 1.F_Complex by COMPLEX1:def 4,COMPLFLD:def 1;
           F is Subring of F_Complex by FIELD_5:12; then
           a * b = 3-CRoot(2) * 1.F_Complex by E1,E2,FIELD_6:16
                .= 3-Root(2);
           hence o in M by E0,ENUMSET1:def 4;
           end;
         suppose E2: b = zeta;
           F is Subring of F_Complex by FIELD_5:12; then
           a * b = 3-CRoot(2) * zeta by E1,E2,FIELD_6:16;
           hence o in M by E0,ENUMSET1:def 4;
           end;
         end;
       suppose E1: a = 3-Root(2)^2;
         per cases by E0,TARSKI:def 2;
         suppose b = 1; then
           E2: b = 1.F_Complex by COMPLEX1:def 4,COMPLFLD:def 1;
           E3: F is Subring of F_Complex by FIELD_5:12;
           E4: 3-Root(2)^2 = 3-Root(2) * 3-Root(2) by O_RING_1:def 1
                          .= 3-CRoot(2) * 3-CRoot(2); then
           a * b = (3-CRoot(2) * 3-CRoot(2)) * 1.F_Complex
                   by E1,E2,E3,FIELD_6:16
                .= 3-Root(2)^2 by E4;
           hence o in M by E0,ENUMSET1:def 4;
           end;
         suppose E2: b = zeta;
           E3: F is Subring of F_Complex by FIELD_5:12;
           E4: 3-Root(2)^2 = 3-Root(2) * 3-Root(2) by O_RING_1:def 1
                          .= 3-CRoot(2) * 3-CRoot(2); then
           a * b = (3-CRoot(2) * 3-CRoot(2)) * zeta by E1,E2,E3,FIELD_6:16
                .= 3-Root(2)^2 * zeta by E4;
           hence o in M by E0,ENUMSET1:def 4;
           end;
         end;
       end;
     hence thesis by E1,TARSKI:2;
     end;
hence thesis by C,FIELD_7:29;
end;
