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 Th47:
  for P being RedSequence of ==>.-relation(TS), k st k in dom P &
  k + 1 in dom P holds ex s, v, t, w st P.k = [s, v] & P.(k + 1) = [t, w]
proof
  let P be RedSequence of ==>.-relation(TS), k;
  assume k in dom P & k + 1 in dom P;
  then [P.k, P.(k + 1)] in ==>.-relation(TS) by REWRITE1:def 2;
  hence thesis by Th31;
end;
