reserve x,y,X for set;

theorem Th29:
  for T being non empty TopSpace, N being net of T for s being
Point of T holds s is_a_cluster_point_of N iff for A being Subset of T,N holds
  s in Cl A
proof
  let T be non empty TopSpace, N be net of T;
  let s be Point of T;
  thus s is_a_cluster_point_of N implies
    for A being Subset of T,N holds s in Cl A by Th8,Th21;
  assume
A1: for A being Subset of T,N holds s in Cl A;
  let V be a_neighborhood of s;
  let i be Element of N;
  reconsider A = rng the mapping of N|i as Subset of T,N by Def2;
  set x = the Element of A /\ Int V;
A2: s in Int V by CONNSP_2:def 1;
  s in Cl A by A1;
  then A meets Int V by A2,PRE_TOPC:def 7;
  then
A3: A /\ Int V <> {}T;
  then
A4: x in Int V by XBOOLE_0:def 4;
A5: Int V c= V by TOPS_1:16;
  x in A by A3,XBOOLE_0:def 4;
  then consider j being object such that
A6: j in dom the mapping of N|i and
A7: x = (the mapping of N|i).j by FUNCT_1:def 3;
A8: the carrier of N|i = {l where l is Element of N: i <= l} by WAYBEL_9:12;
  reconsider j as Element of N|i by A6;
  dom the mapping of N|i = the carrier of N|i by FUNCT_2:def 1;
  then consider l being Element of N such that
A9: j = l and
A10: i <= l by A6,A8;
  take l;
  thus i <= l by A10;
  x = (N|i).j by A7
    .= N.l by A9,WAYBEL_9:16;
  hence thesis by A4,A5;
end;
