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

theorem
  for T being non empty TopSpace,s being sequence of the carrier of T,
  x being Point of T, B being basis of BOOL2F NeighborhoodSystem x holds
  x in lim_f s iff
  for b be Element of B ex i be Nat st
  for j be Nat st i <= j holds s.j in b
  proof
    let T be non empty TopSpace,s be sequence of the carrier of T,
    x be Point of T,
    B be basis of BOOL2F NeighborhoodSystem x;
    hereby
      assume
A1:   x in lim_f s;
      now
        let b be Element of B;
        consider i0 be Element of NAT such that
A2:     for j be Element of NAT st i0 <=j holds s.j in b by A1,Th45;
        reconsider i1=i0 as Nat;
        take i1;
        now
          let k be Nat;
          assume
A3:       i1<=k;
          reconsider k0=k as Element of NAT by ORDINAL1:def 12;
          i0 <= k0 by A3;
          hence s.k in b by A2;
        end;
        hence for k be Nat st i1 <= k holds s.k in b;
      end;
      hence for b be Element of B ex i be Nat st
      for j be Nat st i <=j holds s.j in b;
    end;
    assume
A4: for b be Element of B ex i be Nat st
    for j be Nat st i <= j holds s.j in b;
    now
      let b be Element of B;
      consider i0 be Nat such that
A5:   for j be Nat st i0<=j holds s.j in b by A4;
      reconsider i1=i0 as Element of NAT by ORDINAL1:def 12;
      take i1;
      thus for j be Element of NAT st i1 <= j holds s.j in b by A5;
    end;
    hence x in lim_f s by Th45;
  end;
