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;
