reserve x, x1, x2, y, y1, y2, z, z1, z2 for object, X, X1, X2 for set;
reserve E for non empty set;
reserve e for Element of E;
reserve u, u9, u1, u2, v, v1, v2, w, w1, w2 for Element of E^omega;
reserve F, F1, F2 for Subset of E^omega;
reserve i, k, l, n for Nat;
reserve TS for non empty transition-system over F;
reserve s, s9, s1, s2, t, t1, t2 for Element of TS;
reserve S for Subset of TS;

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;
