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

theorem Th44:
  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
  (for b be Element of B ex i be Element of OrderedNAT st
  for j be Element of OrderedNAT st i <=j holds s.j in b)
  iff
  (for b be Element of B
  ex i be Element of sequence_to_net(s) st
  for j be Element of sequence_to_net(s) st
  i <=j holds (sequence_to_net(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;
A1: (for b be Element of B ex i be Element of OrderedNAT st
    for j be Element of OrderedNAT st i <=j holds s.j in b)
    implies
    (for b be Element of B
    ex i be Element of sequence_to_net(s) st
    for j be Element of sequence_to_net(s) st
    i <=j holds (sequence_to_net(s)).j in b)
    proof
      assume
A2:   for b be Element of B ex i be Element of OrderedNAT st
      for j be Element of OrderedNAT st i <=j holds s.j in b;
      for b be Element of B
      ex i be Element of sequence_to_net(s) st
      for j be Element of sequence_to_net(s) st
      i <=j holds (sequence_to_net(s)).j in b
      proof
        let b be Element of B;
        consider i be Element of OrderedNAT such that
A3:     for j be Element of OrderedNAT st i <=j holds s.j in b by A2;
        reconsider i0=i as Element of sequence_to_net(s);
        for j be Element of sequence_to_net(s) st
        i0 <= j holds (sequence_to_net(s)).j in b
        proof
          let j be Element of sequence_to_net(s);
          assume
A4:       i0<=j;
          reconsider j0=j as Element of OrderedNAT;
          (the mapping of sequence_to_net(s)).j=s.j0 & i<=j0 by A4;
          hence thesis by A3;
        end;
        hence thesis;
      end;
      hence thesis;
    end;
    (for b be Element of B
    ex i be Element of sequence_to_net(s) st
    for j be Element of sequence_to_net(s) st
    i <=j holds (sequence_to_net(s)).j in b)
    implies
    (for b be Element of B ex i be Element of OrderedNAT st
    for j be Element of OrderedNAT st i <=j holds s.j in b)
    proof
      assume
A5: for b be Element of B ex i be Element of sequence_to_net(s) st
      for j be Element of sequence_to_net(s) st
      i <=j holds (sequence_to_net(s)).j in b;
      (for b be Element of B ex i be Element of OrderedNAT st
      for j be Element of OrderedNAT st i <=j holds s.j in b)
      proof
        let b be Element of B;
        consider i be Element of sequence_to_net(s) such that
A6:   for j be Element of sequence_to_net(s) st i <=j holds
        (sequence_to_net(s)).j in b by A5;
        reconsider i0=i as Element of OrderedNAT;
        for j be Element of OrderedNAT st i0 <=j holds s.j in b
        proof
          let j be Element of OrderedNAT;
          assume
A7:       i0 <=j;
          reconsider j0=j as Element of sequence_to_net(s);
          (the mapping of sequence_to_net(s)).j0=s.j & i<=j0 by A7;
          then (sequence_to_net(s)).j0 in b & i <= j0 by A6;
          hence thesis;
        end;
        hence thesis;
      end;
      hence thesis;
    end;
    hence thesis by A1;
  end;
