
theorem ug:
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,T), b1 being Element of E1
st E1 = E & b1 = b holds FAdj(F,{b}\/T) = FAdj(FAdj(F,T),{b1})
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,T), b1 be Element of E1;
assume AS: E1 = E & b1 = b;
A1: F is Subfield of FAdj(FAdj(F,T),{b1}) by FIELD_4:7;
{b} \/ T c= the carrier of FAdj(FAdj(F,T),{b1})
  proof
  now let o be object;
    assume o in {b} \/ T; then
    per cases by XBOOLE_0:def 3;
    suppose o in {b}; then
      B1: o = b by TARSKI:def 1;
      B2: {b1} is Subset of FAdj(FAdj(F,T),{b1}) by FIELD_6:35;
      b in {b1} by AS,TARSKI:def 1;
      hence o in the carrier of FAdj(FAdj(F,T),{b1}) by B1,B2;
      end;
    suppose B1: o in T;
      B2: T is Subset of FAdj(F,T) by FIELD_6:35;
      FAdj(F,T) is Subfield of FAdj(FAdj(F,T),{b1}) by FIELD_6:36; then
      the carrier of FAdj(F,T) c= the carrier of FAdj(FAdj(F,T),{b1})
          by EC_PF_1:def 1;
      hence o in the carrier of FAdj(FAdj(F,T),{b1}) by B2,B1;
      end;
    end;
  hence thesis;
  end; then
A: FAdj(F,{b}\/T) is Subfield of FAdj(FAdj(F,T),{b1}) by AS,A1,FIELD_6:37;
A1: FAdj(F,T) 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
{b1} c= the carrier of FAdj(F,{b}\/T) by AS,XBOOLE_1:11; then
FAdj(FAdj(F,T),{b1}) is Subfield of FAdj(F,{b}\/T) by A1,AS,FIELD_6:37;
hence thesis by A,EC_PF_1:4;
end;
