
theorem
  for X being non empty LinearTopSpace holds
  NeighborhoodSystem 0.X is local_base of X
  proof
    let X be non empty LinearTopSpace;
    reconsider p=0.X as Point of X;
    BOOL2F NeighborhoodSystem 0.X is Subset-Family of X;
    then reconsider NS0 =  (NeighborhoodSystem 0.X) as Subset-Family of X
    by CARDFIL2:def 20;
    for A being a_neighborhood of p ex P being a_neighborhood of p st
    ( P in NeighborhoodSystem p & P c= A ) by YELLOW19:2;
    then NS0 is basis of p by YELLOW13:def 2;
    hence thesis;
  end;
