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;
