
theorem RF2:
for F being Field,
    E being FieldExtension of F,
    T being Subset of E holds RAdj(F,T) = FAdj(F,T) iff RAdj(F,T) is Field
proof
let F be Field, E be FieldExtension of F; let T be Subset of E;
set Pf = FAdj(F,T), Pr = RAdj(F,T);
now assume Pr is Field;
  then reconsider Prf = Pr as Field;
  F is Subring of Prf by RAsub; then
  A: F is Subfield of Prf by FIELD_5:13;
  B: Prf is Subfield of E by FIELD_5:13;
  T is Subset of Prf by RAt; then
  Pf is Subfield of Prf by A,B,FAsub2; then
  C: Pf is Subring of Prf by FIELD_5:12;
  Pr is Subring of Pf by RF;
  hence Pf = Pr by C,RE;
  end;
hence thesis;
end;
