theorem
  not <%>E in rng dom (the Tran of TS) implies for P being RedSequence
of ==>.-relation(TS), k st k in dom P & k + 1 in dom P holds (P.k)`2 <> (P.(k +
  1))`2
proof
  assume
A1: not <%>E in rng dom (the Tran of TS);
  let P be RedSequence of ==>.-relation(TS), k such that
A2: k in dom P & k + 1 in dom P;
  consider s, u, t, v such that
A3: P.k = [s, u] and
A4: P.(k + 1) = [t, v] by A2,Th47;
  [[s, u], [t, v]] in ==>.-relation(TS) by A2,A3,A4,REWRITE1:def 2;
  then u <> v by A1,Th43;
  then (P.k)`2 <> v by A3;
  hence thesis by A4;
end;
