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 Th27:
  ==>.-relation(id (E^omega)) = id (E^omega)
proof
A1: ==>.-relation(id (E^omega)) c= id (E^omega)
  proof
    let x be object;
    assume
A2: x in ==>.-relation(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, 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, id (E^omega);
    [s1, t1] in id (E^omega) by A6;
    then s1 = t1 by RELAT_1:def 10;
    hence thesis by A4,A5,RELAT_1:def 10;
  end;
  id (E^omega) c= ==>.-relation(id (E^omega)) by Th22;
  hence thesis by A1,XBOOLE_0:def 10;
end;
