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
  for P being RedSequence of ==>.-relation(TS), k st k in dom P & k + 1
in dom P holds ex v, w st v = (P.(k + 1))`2 & (P.k)`1, w -->. (P.(k + 1))`1, TS
  & (P.k)`2 = w^v
proof
  let P be RedSequence of ==>.-relation(TS), k such that
A1: k in dom P & k + 1 in dom P;
  consider s, u, t, v such that
A2: P.k = [s, u] and
A3: P.(k + 1) = [t, v] by A1,Th47;
  [[s, u], [t, v]] in ==>.-relation(TS) by A1,A2,A3,REWRITE1:def 2;
  then consider v1, w1 such that
A4: v1 = v and
A5: s, w1 -->. t, TS and
A6: u = w1^v1 by Th35;
  take v1, w1;
  thus v1 = (P.(k + 1))`2 by A3,A4;
  (P.k)`1, w1 -->. t, TS by A2,A5;
  hence thesis by A2,A3,A6;
end;
