theorem Th42:
  X in Rank A iff bool X in Rank succ A
proof
  thus X in Rank A implies bool X in Rank succ A
  proof
    assume
A1: X in Rank A;
 bool X c= Rank A
    by A1,Th41;
    hence thesis by Th32;
  end;
  assume bool X in Rank succ A;
then  X in bool X & bool X c= Rank A by Th32,ZFMISC_1:def 1;
  hence thesis;
end;
