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

theorem
  for T being non empty TopSpace,
  F be Filter of the carrier of T holds
  lim_filter F = lim_filterb F2BOOL(F,T)
  proof
    let T be non empty TopSpace, F be Filter of the carrier of T;
    now
      hereby
        let x be object;assume x in lim_filter F;
        then consider x0 be Point of T such that
A1:     x=x0 and
A2:     F is_filter-finer_than NeighborhoodSystem x0;
        thus x in lim_filterb F2BOOL(F,T) by A1,A2;
      end;
      let x be object;assume x in lim_filterb F2BOOL(F,T);
      then consider x0 be Point of T such that
A3:   x=x0 and
A4:   NeighborhoodSystem x0 c= F;
      F is_filter-finer_than NeighborhoodSystem x0 by A4;
      hence x in lim_filter F by A3;
    end;
    then lim_filter F c= lim_filterb F2BOOL(F,T) &
    lim_filterb F2BOOL(F,T) c= lim_filter F;
    hence thesis;
  end;
