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 Th31:
  ==>.-relation(==>.-relation(S)) = ==>.-relation(S)
proof
A1: ==>.-relation(==>.-relation(S)) c= ==>.-relation(S)
  proof
    let x be object;
    assume
A2: x in ==>.-relation(==>.-relation(S));
    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, ==>.-relation(S) by A2,A4,Def6;
    then a ==>. b, S by Th30;
    hence thesis by A4,Def6;
  end;
  ==>.-relation(S) c= ==>.-relation(==>.-relation(S)) by Th22;
  hence thesis by A1,XBOOLE_0:def 10;
end;
