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 Th28:
  ==>.-relation(S \/ id (E^omega)) = ==>.-relation(S) \/ id (E ^omega)
proof
A1: ==>.-relation(S \/ id (E^omega)) c= ==>.-relation(S) \/ id (E^omega)
  proof
    let x be object;
    assume
A2: x in ==>.-relation(S \/ id (E^omega));
    then consider a, b being object such that
A3: a in E^omega & b in E^omega and
A4: x = [a, b] by ZFMISC_1:def 2;
    reconsider a, b as Element of E^omega by A3;
    a ==>. b, S \/ id (E^omega) by A2,A4,Def6;
    then consider v, w, s1, t1 such that
A5: a = v^s1^w & b = v^t1^w and
A6: s1 -->. t1, S \/ id (E^omega);
A7: [s1, t1] in S \/ id (E^omega) by A6;
    per cases by A7,XBOOLE_0:def 3;
    suppose
      [s1, t1] in S;
      then s1 -->. t1, S;
      then v^s1^w ==>. v^t1^w, S;
      then x in ==>.-relation(S) by A4,A5,Def6;
      hence thesis by XBOOLE_0:def 3;
    end;
    suppose
      [s1, t1] in id (E^omega);
      then s1 = t1 by RELAT_1:def 10;
      then x in id (E^omega) by A4,A5,RELAT_1:def 10;
      hence thesis by XBOOLE_0:def 3;
    end;
  end;
  ==>.-relation(S) \/ id (E^omega) c= ==>.-relation(S \/ id (E^omega))
  proof
    let x be object;
    assume
A8: x in ==>.-relation(S) \/ id (E^omega);
    per cases by A8,XBOOLE_0:def 3;
    suppose
A9:   x in ==>.-relation(S);
      ==>.-relation(S) c= ==>.-relation(S \/ id (E^omega)) by Th26,XBOOLE_1:7;
      hence thesis by A9;
    end;
    suppose
A10:  x in id (E^omega);
A11:  ==>.-relation(id (E^omega)) c= ==>.-relation(S \/ id (E^omega)) by Th26,
XBOOLE_1:7;
      x in ==>.-relation(id (E^omega)) by A10,Th27;
      hence thesis by A11;
    end;
  end;
  hence thesis by A1,XBOOLE_0:def 10;
end;
