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

theorem
  NAT in base_of_frechet_filter
  proof
    reconsider n0=0 as Element of OrderedNAT;
A1: uparrow n0=uparrow {n0};
    uparrow {n0}=NAT
    proof
      NAT c= uparrow {n0}
      proof
        let t be object;
        assume
        t in NAT;
        then reconsider t0=t as Element of OrderedNAT;
        n0 <= t0 & n0 in {n0} by TARSKI:def 1;
        hence thesis by WAYBEL_0:def 16;
      end;
      hence thesis;
    end;
    hence thesis by A1;
  end;
