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

theorem Th34:
  for T being non empty TopSpace, N be net of T, p be Point of T
  st p in Lim N for d being Element of N ex S being Subset of T st S = { N.c
  where c is Element of N : d <= c } & p in Cl S
proof
  let T be non empty TopSpace, N be net of T, p be Point of T such that
A1: p in Lim N;
  let d be Element of N;
  { N.c where c is Element of N : d <= c } c= the carrier of T
  proof
    let x be object;
    assume x in { N.c where c is Element of N : d <= c };
    then ex c being Element of N st x = N.c & d <= c;
    hence thesis;
  end;
  then reconsider S = { N.c where c is Element of N : d <= c } as Subset of T;
  take S;
  thus S = { N.c where c is Element of N : d <= c };
  now
    let G be a_neighborhood of p;
    N is_eventually_in G by A1,Def15;
    then consider i being Element of N such that
A2: for j being Element of N st i <= j holds N.j in G;
    consider e being Element of N such that
A3: d <= e and
A4: i <= e by Def3;
A5: N.e in S by A3;
    N.e in G by A2,A4;
    hence G meets S by A5,XBOOLE_0:3;
  end;
  hence thesis by CONNSP_2:27;
end;
