
theorem RF:
for F being Field,
    E being FieldExtension of F,
    T being Subset of E holds RAdj(F,T) is Subring of FAdj(F,T)
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);
A: 1.Pr = 1.E by dRA .= 1.Pf by dFA;
B: 0.Pr = 0.E by dRA .= 0.Pf by dFA;
now let o be object;
   assume o in the carrier of Pr;
   then o in carrierRA(T) by dRA;
   then consider x being Element of E such that
   B1: x = o & for U being Subring of E
               st F is Subring of U & T is Subset of U holds x in U;
   now let U be Subfield of E;
     assume F is Subfield of U & T is Subset of U;
     then F is Subring of U &
          U is Subring of E & T is Subset of U by RING_3:43;
     hence x in U by B1;
     end;
   then x in carrierFA(T);
   hence o in the carrier of Pf by B1,dFA;
   end;
then C: the carrier of Pr c= the carrier of Pf;
then Y: [:the carrier of Pr,the carrier of Pr:] c=
        [:the carrier of Pf,the carrier of Pf:] by ZFMISC_1:96;
D: (the addF of Pf)||the carrier of Pr
      = ((the addF of E)||carrierFA(T))||the carrier of Pr by dFA
     .= ((the addF of E)||the carrier of Pf)||the carrier of Pr by dFA
     .= (the addF of E)||the carrier of Pr by Y,FUNCT_1:51
     .= (the addF of E)||carrierRA(T) by dRA
     .= the addF of Pr by dRA;
   (the multF of Pf)||the carrier of Pr
      = ((the multF of E)||carrierFA(T))||the carrier of Pr by dFA
     .= ((the multF of E)||the carrier of Pf)||the carrier of Pr by dFA
     .= (the multF of E)||the carrier of Pr by Y,FUNCT_1:51
     .= (the multF of E)||carrierRA(T) by dRA
     .= the multF of Pr by dRA;
  hence thesis by A,B,C,D,C0SP1:def 3;
end;
