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 Th33:
  x in the FinalS of A & x in P implies P in the FinalS of _bool A
proof
  assume x in the FinalS of A & x in P;
  then
A1: P meets the FinalS of A by XBOOLE_0:3;
  P is Element of _bool A by Th16;
  then
  P in { Q where Q is Element of _bool A : Q meets (the FinalS of A) } by A1;
  hence thesis by Def6;
end;
