
theorem ug:
for R being Ring,
    S being RingExtension of R
for T1,T2 being Subset of S
for S1 being RingExtension of RAdj(R,T2), T3 being Subset of S1
st S1 = S & T1 = T3 holds RAdj(R,T1\/T2) = RAdj(RAdj(R,T2),T3)
proof
let F be Ring, E be RingExtension of F; let T1,T2 be Subset of E;
let E1 be RingExtension of RAdj(F,T2), T3 be Subset of E1;
assume AS: E1 = E & T1 = T3;
A1: F is Subring of RAdj(RAdj(F,T2),T3) by FIELD_4:def 1;
T1 \/ T2 c= the carrier of RAdj(RAdj(F,T2),T3)
  proof
  now let o be object;
    assume o in T1 \/ T2; then
    per cases by XBOOLE_0:def 3;
    suppose B1: o in T1;
      T1 is Subset of RAdj(RAdj(F,T2),T3) by AS,FIELD_6:30;
      hence o in the carrier of RAdj(RAdj(F,T2),T3) by B1;
      end;
    suppose B1: o in T2;
      B2: T2 is Subset of RAdj(F,T2) by FIELD_6:30;
      RAdj(F,T2) is Subring of RAdj(RAdj(F,T2),T3) by FIELD_6:31; then
      the carrier of RAdj(F,T2) c= the carrier of RAdj(RAdj(F,T2),T3)
          by C0SP1:def 3;
      hence o in the carrier of RAdj(RAdj(F,T2),T3) by B2,B1;
      end;
    end;
  hence thesis;
  end; then
A: RAdj(F,T1\/T2) is Subring of RAdj(RAdj(F,T2),T3) by AS,A1,FIELD_6:32;
A1: RAdj(F,T2) is Subring of RAdj(F,T1\/T2) by ext0,XBOOLE_1:7;
T2\/T3 is Subset of RAdj(F,T1\/T2) by AS,FIELD_6:30; then
T3 c= the carrier of RAdj(F,T1\/T2) by XBOOLE_1:11; then
RAdj(RAdj(F,T2),T3) is Subring of RAdj(F,T1\/T2) by A1,AS,FIELD_6:32;
hence thesis by A,FIELD_6:12;
end;
