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 Th21:
  s ==>. t, S \/ T implies s ==>. t, S or s ==>. t, T
proof
  assume s ==>. t, S \/ T;
  then consider v, w, s1, t1 such that
A1: s = v^s1^w & t = v^t1^w and
A2: s1 -->. t1, S \/ T;
A3: [s1, t1] in S \/ T by A2;
  per cases by A3,XBOOLE_0:def 3;
  suppose
    [s1, t1] in S;
    then s1 -->. t1, S;
    hence thesis by A1;
  end;
  suppose
    [s1, t1] in T;
    then s1 -->. t1, T;
    hence thesis by A1;
  end;
end;
