
theorem
for F being Field,
    E being FieldExtension of F,
    K being E-extending FieldExtension of F
for T1 being Subset of K,
    T2 being finite Subset of K
st T1 c= T2 & E == FAdj(F,T1) holds FAdj(E,T2) = FAdj(F,T2)
proof
let F be Field, E be FieldExtension of F, K be E-extending FieldExtension of F;
let T1 be Subset of K, T2 be finite Subset of K;
assume AS: T1 c= T2 & E == FAdj(F,T1); then
reconsider T1 as finite Subset of K;
reconsider K1 = K as FieldExtension of FAdj(F,T1) by FIELD_4:7;
reconsider T3 = T2 as Subset of K1;
FAdj(E,T2) = FAdj(FAdj(F,T1),T3) by AS,lemNor3xx .= FAdj(F,T1\/T2) by ug1;
hence thesis by AS,XBOOLE_1:12;
end;
