
theorem
  for X being non empty LinearTopSpace,
      s being sequence of the carrier of X,
      x being Point of X,
      V being local_base of X holds
  x in lim_f s
    iff
  for v being Subset of X st v in (V /\ NeighborhoodSystem 0.X) holds
  ex i being Nat st for j being Nat st i <= j holds s.j in x+v
  proof
    let X be non empty LinearTopSpace,
    s be sequence of the carrier of X,
    x be Point of X, V be local_base of X;
    set B={x+U where U is Subset of X:U in V &
    U is a_neighborhood of 0.X};
    reconsider B as Subset-Family of X by Lm2;
A1: B is basis of BOOL2F NeighborhoodSystem x by Th10;
    hereby
      assume
A2:   x in lim_f s;
      let v be Subset of X;
      assume v in V/\NeighborhoodSystem 0.X;
      then v in V & v in NeighborhoodSystem 0.X by XBOOLE_0: def 4;
      then v in V & v is a_neighborhood of 0.X by YELLOW19:2;
      then x+v in B;
      then reconsider b=x+v as Element of B;
      consider i0 be Nat such that
A3:   for j be Nat st i0 <= j holds s.j in b by A2,A1,CARDFIL2:97;
      thus ex i be Nat st for j be Nat st i <=j holds s.j in x+v by A3;
    end;
    assume
A4: for v be Subset of X st v in (V/\NeighborhoodSystem 0.X) holds
    ex i be Nat st for j be Nat st i <=j holds s.j in x+v;
    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 b be Element of B;
      B is non empty by Lm2;
      then b in B;
      then consider U1 be Subset of X such that
A5:   b=x+U1 and
A6:   U1 in V and
A7:   U1 is a_neighborhood of 0.X;
      U1 in NeighborhoodSystem 0.X by A7,YELLOW19:2;
      then U1 in (V/\NeighborhoodSystem 0.X) by A6,XBOOLE_0:def 4;
      then consider i0 be Nat such that
A8:   for j be Nat st i0 <= j holds s.j in x+U1 by A4;
      take i0;
      thus thesis by A5,A8;
    end;
    hence x in lim_f s by A1,CARDFIL2:97;
  end;
