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 Th30:
  s ==>. t, ==>.-relation(S) implies s ==>. t, S
proof
  assume s ==>. t, ==>.-relation(S);
  then consider v, w, s1, t1 such that
A1: s = v^s1^w & t = v^t1^w and
A2: s1 -->. t1, ==>.-relation(S);
  [s1, t1] in ==>.-relation(S) by A2;
  then s1 ==>. t1, S by Def6;
  hence thesis by A1,Th13;
end;
