reserve
  X for non empty set,
  FX for Filter of X,
  SFX for Subset-Family of X;

theorem Th27:
  #(Tails OrderedNAT) is basis of Frechet_Filter(NAT) &
  Tails_Filter(OrderedNAT)=Frechet_Filter(NAT)
  proof
    consider F be sequence of bool NAT such that
A1: for x be Element of NAT holds F.x =
    {y where y is Element of NAT:x <= y} by Th21;
    set F1=rng F;
    set F2=the set of all uparrow p where p is Element of OrderedNAT;
A2: F1=F2 by A1,Th26;
    hence #(Tails OrderedNAT) is basis of Frechet_Filter(NAT)
    by A1,Th25;
    reconsider BFF=#(Tails OrderedNAT) as
    basis of Frechet_Filter(NAT) by A1,A2,Th25;
    <.#BFF.]=Frechet_Filter(NAT) by Th06;
    hence thesis by DefL9;
  end;
