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};
reserve SA for non empty semiautomaton over Lex(E) \/ {<%>E};
reserve A for non empty automaton over Lex(E) \/ {<%>E};
reserve P for Subset of A;

theorem
  for FA being non empty finite automaton over Lex(E) \/ {<%>E} ex DFA
being non empty deterministic finite automaton over Lex(E) st Lang(FA) = Lang(
  DFA)
proof
  let FA be non empty finite automaton over Lex(E) \/ {<%>E};
  set bNFA = _bool FA;
  Lang(FA) = Lang(bNFA) by Th38;
  hence thesis;
end;
