reserve x for set;
reserve k, l for Nat;
reserve p, q for FinSequence;
reserve R for Relation;
reserve p, q for RedSequence of R;
reserve E for set;
reserve s, t for XFinSequence;
reserve p, q for XFinSequence-yielding FinSequence;
reserve E for set;
reserve S, T, U for semi-Thue-system of E;
reserve s, t, s1, t1, u, v, u1, v1, w for Element of E^omega;
reserve p for FinSequence of E^omega;

theorem
  s ==>. t, S implies S, (S \/ {[s, t]}) are_equivalent_wrt w
proof
  assume s ==>. t, S;
  then [s, t] in ==>.-relation(S) by Def6;
  then {[s, t]} c= ==>.-relation(S) by ZFMISC_1:31;
  then
A1: S \/ {[s, t]} c= S \/ ==>.-relation(S) by XBOOLE_1:9;
  S, S \/ ==>.-relation(S) are_equivalent_wrt w & S c= S \/ {[s, t]} by Th57
,Th60,XBOOLE_1:7;
  hence thesis by A1,Th56;
end;
