reserve W,X,Y,Z for set,
  f,g for Function,
  a,x,y,z for set;
reserve u,v for Element of Tarski-Class(X),
  A,B,C for Ordinal,
  L for Sequence;

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;
