theorem Th65:
  X c= Rank A iff the_rank_of X c= A
proof
  thus X c= Rank A implies the_rank_of X c= A by Def9;
  assume the_rank_of X c= A;
then A1: Rank the_rank_of X c= Rank A by Th37;
 X c= Rank the_rank_of X by Def9;
  hence thesis by A1;
end;
