theorem Th35:
  omega in W implies ex a,M st a is_cofinal_with omega & M = Rank
  a & M is_elementary_subsystem_of W
proof
  set a = the Ordinal of W;
  assume
A1: omega in W;
  then consider phi such that
A2: phi is increasing & phi is continuous and
A3: for a,M st phi.a = a & {} <> a & M = Rank a holds M
  is_elementary_subsystem_of W by Th33;
  consider b such that
A4: a in b and
A5: phi.b = b & b is_cofinal_with omega by A1,A2,Th29;
  reconsider M = Rank b as non empty set by A4,CLASSES1:36;
  take b,M;
  thus thesis by A3,A4,A5;
end;
