
theorem RAsub:
for R being Ring,
    S being RingExtension of R
for T being Subset of S holds R is Subring of RAdj(R,T)
proof
let R be Ring, S be RingExtension of R; let T be Subset of S;
set P = RAdj(R,T);
X: R is Subring of S by FIELD_4:def 1;
A: 1.P = 1.S by dRA .= 1.R by X,C0SP1:def 3;
B: 0.P = 0.S by dRA .= 0.R by X,C0SP1:def 3;
C: the carrier of R c= the carrier of P
   proof
   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 Subring of S;
       assume R is Subring of U & T is Subset of U;
       then the carrier of R c= the carrier of U by C0SP1:def 3;
       hence o in U by C1;
       end;
     then a in carrierRA(T);
     hence o in the carrier of P by dRA;
     end;
   hence thesis;
   end;
then Y: [:the carrier of R,the carrier of R:] c=
        [:the carrier of P,the carrier of P:] by ZFMISC_1:96;
D: (the addF of P)||the carrier of R
      = ((the addF of S)||carrierRA(T))||the carrier of R by dRA
     .= ((the addF of S)||the carrier of P)||the carrier of R by dRA
     .= (the addF of S)||the carrier of R by Y,FUNCT_1:51
     .= the addF of R by X,C0SP1:def 3;
   (the multF of P)||the carrier of R
      = ((the multF of S)||carrierRA(T))||the carrier of R by dRA
     .= ((the multF of S)||the carrier of P)||the carrier of R by dRA
     .= (the multF of S)||the carrier of R by Y,FUNCT_1:51
     .= the multF of R by X,C0SP1:def 3;
hence thesis by A,B,C,D,C0SP1:def 3;
end;
