
theorem ug1:
for F being Field,
    E being FieldExtension of F
for T1 being Subset of E, T2 being Subset of E
for E1 being FieldExtension of FAdj(F,T2), T3 being Subset of E1
st E1 = E & T1 = T3 holds FAdj(F,T1\/T2) = FAdj(FAdj(F,T2),T3)
proof
let F be Field, E be FieldExtension of F;
let T1 be Subset of E, T2 be Subset of E;
let E1 be FieldExtension of FAdj(F,T2), T3 be Subset of E1;
assume AS: E1 = E & T1 = T3;
A1: F is Subfield of FAdj(FAdj(F,T2),T3) by FIELD_4:7;
T1 \/ T2 c= the carrier of FAdj(FAdj(F,T2),T3)
  proof
  now let o be object;
    assume o in T1 \/ T2; then
    per cases by XBOOLE_0:def 3;
    suppose B1: o in T1;
      T1 is Subset of FAdj(FAdj(F,T2),T3) by AS,FIELD_6:35;
      hence o in the carrier of FAdj(FAdj(F,T2),T3) by B1;
      end;
    suppose B1: o in T2;
      B2: T2 is Subset of FAdj(F,T2) by FIELD_6:35;
      FAdj(F,T2) is Subfield of FAdj(FAdj(F,T2),T3) by FIELD_6:36; then
      the carrier of FAdj(F,T2) c= the carrier of FAdj(FAdj(F,T2),T3)
          by EC_PF_1:def 1;
      hence o in the carrier of FAdj(FAdj(F,T2),T3) by B2,B1;
      end;
    end;
  hence thesis;
  end; then
A: FAdj(F,T1\/T2) is Subfield of FAdj(FAdj(F,T2),T3) by AS,A1,FIELD_6:37;
A1: FAdj(F,T2) is Subfield of FAdj(F,T1\/T2) by FIELD_7:10,XBOOLE_1:7;
T2\/T3 is Subset of FAdj(F,T1\/T2) by AS,FIELD_6:35; then
T3 c= the carrier of FAdj(F,T1\/T2) by XBOOLE_1:11; then
FAdj(FAdj(F,T2),T3) is Subfield of FAdj(F,T1\/T2) by A1,AS,FIELD_6:37;
hence thesis by A,EC_PF_1:4;
end;
