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;
