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
  X c= Y implies the_rank_of X c= the_rank_of Y
proof
  assume
A1: X c= Y;
 Y c= Rank the_rank_of Y by Def9;
then  X c= Rank the_rank_of Y by A1;
  hence thesis by Def9;
end;
