
theorem
Roots(F_Real,X^3-2) = {3-Root(2)}
proof
H0: F_Real is Subfield of F_Complex by FIELD_4:7;
H1: Roots(F_Real,X^3-2) = {a where a is Element of F_Real :
                           a is_a_root_of X^3-2,F_Real} by FIELD_4:def 4;
H2: Roots(F_Complex,X^3-2) = {a where a is Element of F_Complex :
                              a is_a_root_of X^3-2,F_Complex} by FIELD_4:def 4;
A: now let o be object;
   assume o in {3-Root(2)}; then
   A1: o = 3-Root(2) by TARSKI:def 1;
   3-Root(2) is_a_root_of X^3-2,F_Real by LL2,FIELD_4:def 2;
   hence o in Roots(F_Real,X^3-2) by A1,H1;
   end;
now let o be object;
  assume o in Roots(F_Real,X^3-2); then
  consider a being Element of F_Real such that
  B1: o = a & a is_a_root_of X^3-2,F_Real by H1;
  the carrier of F_Real c= the carrier of F_Complex by H0,EC_PF_1:def 1; then
  reconsider z = a as Element of F_Complex;
  0.F_Real = Ext_eval(X^3-2,a) by B1,FIELD_4:def 2
          .= Ext_eval(X^3-2,z) by FIELD_6:11; then
  0.F_Complex = Ext_eval(X^3-2,z) by H0,EC_PF_1:def 1; then
  z is_a_root_of X^3-2,F_Complex by FIELD_4:def 2; then
  B2: z in Roots(F_Complex,X^3-2) by H2;
  now assume z <> 3-Root(2); then
    per cases by B2,lemroots,ENUMSET1:def 1;
    suppose z = 3-Root(2) * zeta;
      hence contradiction by lemnotsplit;
      end;
    suppose z = 3-Root(2) * zeta^2;
      hence contradiction by lemnotsplit;
      end;
    end;
  hence o in {3-Root(2)} by B1,TARSKI:def 1;
  end;
hence thesis by A,TARSKI:2;
end;
