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 Th34:
  X in the FinalS of _bool A implies X meets the FinalS of A
proof
  assume
A1: X in the FinalS of _bool A;
  the FinalS of _bool A = { Q where Q is Element of _bool A : Q meets (the
  FinalS of A) } by Def6;
  then
  ex Q being Element of _bool A st X = Q & Q meets (the FinalS of A) by A1;
  hence thesis;
end;
