reserve i for Nat,
  j for Element of NAT,
  X,Y,x,y,z for set;

theorem Th21:
  the_rank_of varcl X = the_rank_of X
proof
A1: X c= Rank the_rank_of X by CLASSES1:def 9;
  set a = the_rank_of X;
A2: a c= succ a by ORDINAL3:1;
  succ a c= succ succ a by ORDINAL3:1;
  then a c= succ succ a by A2;
  then
A3: Rank a c= Rank succ succ a by CLASSES1:37;
  now
    let x,y;
    assume [x,y] in Rank the_rank_of X;
    then x in Rank a by A3,CLASSES1:45;
    hence x c= Rank the_rank_of X by ORDINAL1:def 2;
  end;
  then varcl X c= Rank a by A1,Def1;
  hence the_rank_of varcl X c= a by CLASSES1:65;
  X c= varcl X by Def1;
  hence thesis by CLASSES1:67;
end;
