theorem Th39:
  omega in W implies ex a,M st a is_cofinal_with omega & M = Rank a & M <==> W
proof
  assume omega in W;
  then consider b,M such that
A1: b is_cofinal_with omega & M = Rank b & M is_elementary_subsystem_of
  W by Th35;
  take b,M;
  thus thesis by A1,Th9;
end;
