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

theorem Th45:
  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 Element of NAT st
  for j be Element of 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;
    now
      hereby
        assume
A1:     x in lim_filter(s,Frechet_Filter(NAT));
        now
          let b be Element of B;
          consider i0 be Element of OrderedNAT such that
A2:       for j be Element of OrderedNAT st i0 <= j holds s.j in b
          by A1,Th41;
          reconsider i1=i0 as Element of NAT;
          now
            let j be Element of NAT;
            assume
A3:         i1<=j;
            reconsider j1=j as Element of OrderedNAT;
            i0<=j1 by A3;
            hence s.j in b by A2;
          end;
          hence ex i be Element of NAT st
          for j be Element of NAT st i <=j holds s.j in b;
        end;
        hence for b be Element of B ex i be Element of NAT st
        for j be Element of NAT st i <=j holds s.j in b;
      end;
      assume
A4:   for b be Element of B ex i be Element of NAT st
      for j be Element of NAT st i <=j holds s.j in b;
      now
        let b be Element of B;
        consider i0 be Element of NAT such that
A5:     for j be Element of NAT st i0<=j holds s.j in b by A4;
        reconsider i1=i0 as Element of OrderedNAT;
        now
          let j be Element of OrderedNAT;
          assume
A6:       i1 <= j;
          consider x,y be Element of NAT such that
A7:       [i1,j]=[x,y] and
A8:       x<=y by A6;
          i1=x & j=y & x<=y by A7,A8,XTUPLE_0:1;
          hence s.j in b by A5;
        end;
        hence ex i be Element of OrderedNAT st
        for j be Element of OrderedNAT st i <=j holds s.j in b;
      end;
      hence x in lim_filter(s,Frechet_Filter(NAT)) by Th41;
    end;
    hence thesis;
  end;
