
theorem
for R being strict Ring,
    S being RingExtension of R,
    T being Subset of S holds RAdj(R,T) = R iff T is Subset of R
proof
let R be strict Ring, S be RingExtension of R; let T be Subset of S;
set P = RAdj(R,T);
X: R is Subring of R & R is Subring of S by FIELD_4:def 1,LIOUVIL2:18;
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 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;
now assume B0: T is Subset of R;
  B5: 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
      B1: 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 R by X,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 dRA .= 1.R by X,C0SP1:def 3;
  B3: 0.P = 0.S by dRA .= 0.R by X,C0SP1:def 3;
  B4: the addF of P = (the addF of S)||carrierRA(T) by dRA
                   .= (the addF of S)||the carrier of R by B1,dRA
                   .= the addF of R by X,C0SP1:def 3;
  the multF of P = (the multF of S)||carrierRA(T) by dRA
                .= (the multF of S)||the carrier of R by B1,dRA
                .= the multF of R by X,C0SP1:def 3;
  hence RAdj(R,T) = R by Z,TARSKI:2,B5,B2,B3,B4;
  end;
hence thesis by RAt;
end;
