
theorem FAsub2:
for F being Field,
    E being FieldExtension of F,
    T being Subset of E
for U being Subfield of E st F is Subfield of U & T is Subset of U
holds FAdj(F,T) is Subfield of U
proof
let R be Field, S be FieldExtension of R; let T be Subset of S;
let U be Subfield of S;
assume AS: R is Subfield of U & T is Subset of U;
set P = FAdj(R,T);
A: 1.P = 1.S by dFA .= 1.U by EC_PF_1:def 1;
B: 0.P = 0.S by dFA .= 0.U by EC_PF_1:def 1;
C: the carrier of P c= the carrier of U
   proof
   now let o be object;
     assume o in the carrier of P;
     then o in carrierFA(T) by dFA;
     then consider x being Element of S such that
     C2: x = o & for U being Subfield of S
                 st R is Subfield of U & T is Subset of U holds x in U;
     x in U by C2,AS;
     hence o in the carrier of U by C2;
     end;
   hence thesis;
   end;
then Y: [:the carrier of P,the carrier of P:] c=
        [:the carrier of U,the carrier of U:] by ZFMISC_1:96;
D: (the addF of U)||the carrier of P
      = ((the addF of S)||the carrier of U)||the carrier of P by EC_PF_1:def 1
     .= (the addF of S)||the carrier of P by Y,FUNCT_1:51
     .= (the addF of S)||carrierFA(T) by dFA
     .= the addF of P by dFA;
   (the multF of U)||the carrier of P
      = ((the multF of S)||the carrier of U)||the carrier of P by EC_PF_1:def 1
     .= (the multF of S)||the carrier of P by Y,FUNCT_1:51
     .= (the multF of S)||carrierFA(T) by dFA
     .= the multF of P by dFA;
hence thesis by A,B,C,D,EC_PF_1:def 1;
end;
