
theorem Th26:
  for T being non empty TopSpace for N being net of T for S being
  Subset of T st N is_eventually_in S holds Lim N c= Cl S
proof
  let T be non empty TopSpace, N be net of T, S be Subset of T;
  given i being Element of N such that
A1: for j being Element of N st j >= i holds N.j in S;
  let x be object;
  assume
A2: x in Lim N;
  then reconsider x as Element of T;
  now
    let G be a_neighborhood of x;
    N is_eventually_in G by A2,YELLOW_6:def 15;
    then consider k being Element of N such that
A3: for j being Element of N st j >= k holds N.j in G;
    [#]N is directed by WAYBEL_0:def 6;
    then consider j being Element of N such that
    j in [#]N and
A4: j >= i and
A5: j >= k;
A6: N.j in G by A3,A5;
    N.j in S by A1,A4;
    hence G meets S by A6,XBOOLE_0:3;
  end;
  hence thesis by CONNSP_2:27;
end;
