
theorem ext1:
for F being Field,
    E being FieldExtension of F
for a being Element of E holds FAdj(F,{a,-a}) = FAdj(F,{a})
proof
let F be Field, E be FieldExtension of F; let a be Element of E;
Y: FAdj(F,{a}) is Subfield of FAdj(F,{a,-a}) by FIELD_7:10,ZFMISC_1:7;
now let o be object;
  assume A: o in the carrier of FAdj(F,{a,-a});
  the carrier of FAdj(F,{a,-a}) = carrierFA({a,-a}) by FIELD_6:def 6
  .= {x where x is Element of E :
       for U being Subfield of E st F is Subfield of U &
                        {a,-a} is Subset of U holds x in U};
  then consider oe being Element of E such that
  C: oe = o &
     for U being Subfield of E st F is Subfield of U &
                        {a,-a} is Subset of U holds oe in U by A;
  D: now let U be Subfield of E;
     assume D: F is Subfield of U & {a} is Subset of U;
     E: a in {a} by TARSKI:def 1; then
     reconsider au = a as Element of U by D;
     U is Subring of E by FIELD_5:12; then
     F: -au = -a by FIELD_6:17;
     now let u be object;
       assume u in {a,-a};
       then per cases by TARSKI:def 2;
       suppose u = a;
         hence u in the carrier of U by E,D;
         end;
       suppose u = -a;
         hence u in the carrier of U by F;
         end;
       end;
    then {a,-a} is Subset of U by TARSKI:def 3;
    hence oe in U by C,D;
    end;
  the carrier of FAdj(F,{a}) = carrierFA({a}) by FIELD_6:def 6
  .= {x where x is Element of E :
       for U being Subfield of E st F is Subfield of U &
                        {a} is Subset of U holds x in U};
  hence o in the carrier of FAdj(F,{a}) by C,D;
  end; then
the carrier of FAdj(F,{a,-a}) c= the carrier of FAdj(F,{a}); then
FAdj(F,{a,-a}) is Subfield of FAdj(F,{a}) by EC_PF_1:6;
hence thesis by Y,EC_PF_1:4;
end;
