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 Th12:
  s ==>. t, S implies u^s ==>. u^t, S & s^u ==>. t^u, S
proof
  given v, w, s1, t1 such that
A1: s = v^s1^w and
A2: t = v^t1^w and
A3: s1 -->. t1, S;
A4: u^t = u^(v^t1)^w by A2,AFINSQ_1:27
    .= u^v^t1^w by AFINSQ_1:27;
  u^s = u^(v^s1)^w by A1,AFINSQ_1:27
    .= u^v^s1^w by AFINSQ_1:27;
  hence u^s ==>. u^t, S by A3,A4;
  s^u = v^s1^(w^u) & t^u = v^t1^(w^u) by A1,A2,AFINSQ_1:27;
  hence thesis by A3;
end;
