reserve x,y,z,X for set,
  T for Universe;

theorem
  for T being non empty TopSpace, N being net of T for p being Point of
  T st not p in Lim N ex Y being subnet of N st not ex Z being subnet of Y st p
  in Lim Z
proof
  let T be non empty TopSpace, N be net of T;
  let p be Point of T;
  assume not p in Lim N;
  then consider V be a_neighborhood of p such that
A1: not N is_eventually_in V by Def15;
  N is_often_in (the carrier of T) \ V by A1,WAYBEL_0:9;
  then reconsider Y = N"((the carrier of T) \ V) as subnet of N by Th22;
  take Y;
  let Z be subnet of Y;
  assume p in Lim Z;
  then Z is_eventually_in V by Def15;
  then
A2: Y is_often_in V by Th18,Th19;
  Y is_eventually_in (the carrier of T) \ V by Th23;
  hence contradiction by A2,WAYBEL_0:10;
end;
