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 Th25:
  S is Thue-system of E implies ==>.-relation(S) = (==>.-relation( S))~
proof
  assume
A1: S is Thue-system of E;
  now
    let x be object;
    thus x in ==>.-relation(S) implies x in (==>.-relation(S))~
    proof
      assume
A2:   x in ==>.-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, S by A2,A4,Def6;
      then b ==>. a, S by A1,Th17;
      then [b, a] in ==>.-relation(S) by Def6;
      hence thesis by A4,RELAT_1:def 7;
    end;
    thus x in (==>.-relation(S))~ implies x in ==>.-relation(S)
    proof
      assume
A5:   x in (==>.-relation(S))~;
      then consider a, b being object such that
A6:   a in E^omega & b in E^omega and
A7:   x = [a, b] by ZFMISC_1:def 2;
      reconsider a, b as Element of E^omega by A6;
      [b, a] in ==>.-relation(S) by A5,A7,RELAT_1:def 7;
      then b ==>. a, S by Def6;
      then a ==>. b, S by A1,Th17;
      hence thesis by A7,Def6;
    end;
  end;
  hence thesis by TARSKI:2;
end;
