
theorem ext0:
for R being Ring,
    S being RingExtension of R
for T1,T2 being Subset of S
st T1 c= T2 holds RAdj(R,T1) is Subring of RAdj(R,T2)
proof
let F be Ring, E be RingExtension of F; let T1,T2 be Subset of E;
assume AS: T1 c= T2;
H: T2 is Subset of RAdj(F,T2) by FIELD_6:30;
now let o be object;
  assume o in T1;
  then o in T2 by AS;
  hence o in the carrier of RAdj(F,T2) by H;
  end; then
A: T1 c= the carrier of RAdj(F,T2);
F is Subring of RAdj(F,T2) by FIELD_4:def 1;
hence RAdj(F,T1) is Subring of RAdj(F,T2) by A,FIELD_6:32;
end;
