
theorem
  for T being non empty TopSpace for x,y being Element of InclPoset the
topology of T st x << y for F being ultra Filter of BoolePoset the carrier of T
  st x in F ex p being Element of T st p in y & p is_a_convergence_point_of F,T
proof
  let T be non empty TopSpace;
  let x,y be Element of InclPoset the topology of T such that
A1: x << y;
  let F be ultra Filter of BoolePoset the carrier of T;
  assume x in F;
  then consider p being Element of T such that
A2: p in y & p is_a_cluster_point_of F,T by A1,Th28;
  take p;
  thus thesis by A2,Th27;
end;
