theorem
  the_rank_of X = {} iff X = {}
proof
  thus the_rank_of X = {} implies X = {}
  proof
    assume the_rank_of X = {};
then  X c= Rank {} by Def9;
    hence thesis by Lm2;
  end;
  assume X = {};
then  for Y st Y in X holds the_rank_of Y in {};
  hence the_rank_of X c= {} by Th69;
  thus {} c= the_rank_of X;
end;
