
theorem ug1:
for F being Field,
    E being FieldExtension of F
for b being Element of E, T being Subset of E
for E1 being FieldExtension of FAdj(F,{b}), T1 being Subset of E1
st E1 = E & T1 = T holds FAdj(F,{b}\/T) = FAdj(FAdj(F,{b}),T1)
proof
let F be Field, E be FieldExtension of F;
let b be Element of E, T be Subset of E;
let E1 be FieldExtension of FAdj(F,{b}), T1 be Subset of E1;
assume AS: E1 = E & T1 = T;
A1: F is Subfield of FAdj(FAdj(F,{b}),T1) by FIELD_4:7;
{b} \/ T c= the carrier of FAdj(FAdj(F,{b}),T1)
  proof
  now let o be object;
    assume o in {b} \/ T; then
    per cases by XBOOLE_0:def 3;
    suppose B1: o in {b};
      {b} is Subset of FAdj(FAdj(F,{b}),T1)
          proof
          now let o be object;
            assume B3: o in {b};
            B4: {b} is Subset of FAdj(F,{b}) by FIELD_6:35;
            FAdj(F,{b}) is Subfield of FAdj(FAdj(F,{b}),T1) by FIELD_6:36;
            then the carrier of FAdj(F,{b}) c=
                 the carrier of FAdj(FAdj(F,{b}),T1) by EC_PF_1:def 1;
            hence o in the carrier of FAdj(FAdj(F,{b}),T1) by B3,B4;
            end;
          hence thesis by TARSKI:def 3;
          end;
      hence o in the carrier of FAdj(FAdj(F,{b}),T1) by B1;
      end;
    suppose B1: o in T;
      T is Subset of FAdj(FAdj(F,{b}),T1) by AS,FIELD_6:35;
      hence o in the carrier of FAdj(FAdj(F,{b}),T1) by B1;
      end;
    end;
  hence thesis;
  end; then
A: FAdj(F,{b}\/T) is Subfield of FAdj(FAdj(F,{b}),T1) by AS,A1,FIELD_6:37;
A1: FAdj(F,{b}) is Subfield of FAdj(F,{b}\/T) by ext0,XBOOLE_1:7;
{b}\/T is Subset of FAdj(F,{b}\/T) by FIELD_6:35; then
T1 c= the carrier of FAdj(F,{b}\/T) by AS,XBOOLE_1:11; then
FAdj(FAdj(F,{b}),T1) is Subfield of FAdj(F,{b}\/T) by A1,AS,FIELD_6:37;
hence thesis by A,EC_PF_1:4;
end;
