reserve x, y, X for set;
reserve E for non empty set;
reserve e for Element of E;
reserve u, u1, v, v1, v2, w, w9, w1, w2 for Element of E^omega;
reserve F for Subset of E^omega;
reserve i, k, l for Nat;
reserve TS for non empty transition-system over F;
reserve S, T for Subset of TS;
reserve SA for non empty semiautomaton over F;
reserve A for non empty automaton over F;
reserve p, q for Element of A;
reserve TS for non empty transition-system over Lex(E) \/ {<%>E};

theorem Th27:
  x, u^w ==>* y, v^w, TS implies x, u ==>* y, v, TS
proof
  assume x, u^w ==>* y, v^w, TS;
  then ==>.-relation(TS) reduces [x, u^w], [y, v^w] by REWRITE3:def 6;
  then ==>.-relation(TS) reduces [x, u], [y, v] by Th26;
  hence thesis by REWRITE3:def 6;
end;
