reserve x,y,X for set;

theorem Th11:
  for T being non empty TopSpace for N being net of T for x being
Point of T holds x is_a_cluster_point_of N iff x is_a_cluster_point_of a_filter
  N, T
proof
  let T be non empty TopSpace;
  let N be net of T;
  set F = a_filter N;
  let x be Point of T;
  thus x is_a_cluster_point_of N implies x is_a_cluster_point_of F, T
  proof
    assume
A1: for O being a_neighborhood of x holds N is_often_in O;
    let A be Subset of T;
    assume that
A2: A is open and
A3: x in A;
    let B be set;
    assume B in F;
    then N is_eventually_in B by Th10;
    then consider i being Element of N such that
A4: for j being Element of N st i <= j holds N.j in B;
    A is a_neighborhood of x by A2,A3,CONNSP_2:3;
    then N is_often_in A by A1;
    then ex j being Element of N st i <= j & N.j in A;
    then ex a being Point of T st a in B & a in A by A4;
    hence thesis by XBOOLE_0:3;
  end;
  assume
A5: for A being Subset of T st A is open & x in A for B being set st B
  in F holds A meets B;
  let O be a_neighborhood of x;
  let i be Element of N;
  reconsider B = rng the mapping of N|i as Subset of T,N by Def2;
  N is_eventually_in B by Th8;
  then
A6: B in F;
  x in Int O by CONNSP_2:def 1;
  then Int O meets B by A5,A6;
  then consider x being object such that
A7: x in Int O and
A8: x in B by XBOOLE_0:3;
  consider j being Element of N such that
A9: i <= j and
A10: x = N.j by A8,Th7;
  take j;
  Int O c= O by TOPS_1:16;
  hence thesis by A7,A9,A10;
end;
