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 Th49:
  for P being RedSequence of ==>.-relation(TS), k st k in dom P &
k + 1 in dom P holds (P.k)`1 in TS & (P.k)`2 in E^omega & (P.(k + 1))`1 in TS &
(P.(k + 1))`2 in E^omega & (P.k)`1 in dom dom (the Tran of TS) & (P.(k + 1))`1
  in rng (the Tran of TS)
proof
  let P be RedSequence of ==>.-relation(TS), k;
  assume k in dom P & k + 1 in dom P;
  then
A1: [P.k, P.(k + 1)] in ==>.-relation(TS) by REWRITE1:def 2;
  then consider s, v, t, w such that
A2: P.k = [s, v] & P.(k + 1) = [t, w] by Th31;
A3: s in dom dom (the Tran of TS) & t in rng (the Tran of TS) by A1,A2,Th32;
  thus thesis by A2,A3;
end;
