theorem
  omega in W implies ex phi st phi is increasing & phi is continuous &
  for a,M st phi.a = a & {} <> a & M = Rank a holds M <==> W
proof
  assume omega in W;
  then consider phi such that
A1: phi is increasing & phi is continuous and
A2: for a,M st phi.a = a & {} <> a & M = Rank a holds M
  is_elementary_subsystem_of W by Th33;
  take phi;
  thus phi is increasing & phi is continuous by A1;
  let a,M;
  assume phi.a = a & {} <> a & M = Rank a;
  hence thesis by A2,Th9;
end;
