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;
reserve n for Element of omega;

theorem Th66:
  X in Rank A iff the_rank_of X in A
proof
  thus X in Rank A implies the_rank_of X in A
  proof
    assume X in Rank A;
then  bool X in Rank succ A by Th42;
then A1: bool X c= Rank A by Th32;
 the_rank_of bool X = succ the_rank_of X by Th63;
then A2: the_rank_of X in the_rank_of bool X by ORDINAL1:6;
 the_rank_of bool X c= A by A1,Def9;
    hence thesis by A2;
  end;
  assume
A3: the_rank_of X in A;
 X c= Rank the_rank_of X by Def9;
then A4: X in Rank succ the_rank_of X by Th32;
 Rank succ the_rank_of X c= Rank A by A3,Th37,ORDINAL1:21;
  hence thesis by A4;
end;
