
theorem RAsub2:
for R being Ring,
    S being RingExtension of R,
    T being Subset of S
for U being Subring of S st R is Subring of U & T is Subset of U
holds RAdj(R,T) is Subring of U
proof
let R be Ring, S be RingExtension of R; let T be Subset of S;
let U be Subring of S;
assume AS: R is Subring of U & T is Subset of U;
set P = RAdj(R,T);
A: 1.P = 1.S by dRA .= 1.U by C0SP1:def 3;
B: 0.P = 0.S by dRA .= 0.U by C0SP1:def 3;
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 carrierRA(T) by dRA;
     then consider x being Element of S such that
     C2: x = o & for U being Subring of S
                 st R is Subring 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 C0SP1:def 3
     .= (the addF of S)||the carrier of P by Y,FUNCT_1:51
     .= (the addF of S)||carrierRA(T) by dRA
     .= the addF of P by dRA;
    (the multF of U)||the carrier of P
      = ((the multF of S)||the carrier of U)||the carrier of P by C0SP1:def 3
     .= (the multF of S)||the carrier of P by Y,FUNCT_1:51
     .= (the multF of S)||carrierRA(T) by dRA
     .= the multF of P by dRA;
hence thesis by A,B,C,D,C0SP1:def 3;
end;
