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 ==>. s, S implies ex v, w, s1 st s = v^s1^w & s1 -->. s1, S
proof
  given v, w, s1, t1 such that
A1: s = v^s1^w and
A2: s = v^t1^w and
A3: s1 -->. t1, S;
  v^s1 = v^t1 by A1,A2,AFINSQ_1:28;
  then s1 = t1 by AFINSQ_1:28;
  hence thesis by A1,A3;
end;
