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 Th30:
  _bool TS is non empty transition-system over Lex(E) \/ {<%>E}
proof
  Lex(E) c= Lex(E) \/ {<%>E} by XBOOLE_1:7;
  then dom the Tran of _bool TS c= [: the carrier of _bool TS, Lex(E) :] & [:
  the carrier of _bool TS, Lex(E) :] c= [: the carrier of _bool TS, Lex(E) \/ {
  <%>E} :] by ZFMISC_1:95;
  hence thesis by RELSET_1:5,XBOOLE_1:1;
end;
