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 is Thue-system of E & s ==>* t, S implies t ==>* s, S
proof
  assume that
A1: S is Thue-system of E and
A2: s ==>* t, S;
  ==>.-relation(S) reduces s, t by A2;
  then consider p being RedSequence of ==>.-relation(S) such that
A3: p.1 = s and
A4: p.(len p) = t by REWRITE1:def 3;
  set q = Rev p;
  q.(len p) = s by A3,FINSEQ_5:62;
  then
A5: q.(len q) = s by FINSEQ_5:def 3;
  q is RedSequence of (==>.-relation(S))~ by REWRITE1:9;
  then
A6: q is RedSequence of ==>.-relation(S) by A1,Th25;
  q.1 = t by A4,FINSEQ_5:62;
  then ==>.-relation(S) reduces t, s by A6,A5,REWRITE1:def 3;
  hence thesis;
end;
