
theorem Th1:
for F being Field
for E being FieldExtension of F
for T being Subset of E holds FAdj(F,T) == F iff T is Subset of F
proof
let R be Field, S be FieldExtension of R; let T be Subset of S;
set P = FAdj(R,T);
X: R is Subring of S by FIELD_4:def 1;
X1: R is Subfield of R & R is Subfield of S by FIELD_4:7,FIELD_4:1;
X2: carrierFA(T) = {x where x is Element of S :
      for U being Subfield of S
               st R is Subfield of U & T is Subset of U holds x in U}
    by FIELD_6:def 5;
Z: now let o be object;
   assume C1: o in the carrier of R;
   the carrier of R c= the carrier of S by X,C0SP1:def 3;
   then reconsider a = o as Element of S by C1;
   now let U be Subfield of S;
     assume R is Subfield of U & T is Subset of U;
     then the carrier of R c= the carrier of U by EC_PF_1:def 1;
     hence o in U by C1;
     end;
   then a in carrierFA(T) by X2;
   hence o in the carrier of P by FIELD_6:def 6;
   end;
now assume B0: T is Subset of R;
  B5: now let o be object;
      assume o in the carrier of P;
      then o in carrierFA(T) by FIELD_6:def 6;
      then consider x being Element of S such that
      B1: x = o & for U being Subfield of S
                  st R is Subfield of U & T is Subset of U holds x in U
          by X2;
      x in R by X1,B0,B1;
      hence o in the carrier of R by B1;
      end; then
  B1: the carrier of R = the carrier of P by Z,TARSKI:2;
  B2: 1.P = 1.S by FIELD_6:def 6 .= 1.R by X,C0SP1:def 3;
  B3: 0.P = 0.S by FIELD_6:def 6 .= 0.R by X,C0SP1:def 3;
  B4: the addF of P = (the addF of S)||carrierFA(T) by FIELD_6:def 6
                   .= (the addF of S)||the carrier of R by B1,FIELD_6:def 6
                   .= (the addF of R)||the carrier of R by X,C0SP1:def 3;
  the multF of P = (the multF of S)||carrierFA(T) by FIELD_6:def 6
                .= (the multF of S)||the carrier of R by B1,FIELD_6:def 6
                .= (the multF of R)||the carrier of R by X,C0SP1:def 3;
  hence FAdj(R,T) == R by B5,B2,B3,B4,Z,TARSKI:2;
  end;
hence thesis by FIELD_6:35;
end;
