reserve x,y,X for set;

theorem
  for T being non empty TopSpace, A being Subset of T holds A is closed
  iff for N being net of T st N is_eventually_in A for x being Point of T st x
  is_a_cluster_point_of N holds x in A
proof
  let T be non empty TopSpace, A be Subset of T;
  A is closed iff Cl A = A by PRE_TOPC:22;
  hence A is closed implies for N being net of T st N is_eventually_in A for x
  being Point of T st x is_a_cluster_point_of N holds x in A by Th21;
  assume
A1: for N being net of T st N is_eventually_in A for x being Point of T
  st x is_a_cluster_point_of N holds x in A;
  A = Cl A
  proof
    thus A c= Cl A by PRE_TOPC:18;
    let x be object;
    assume
A2: x in Cl A;
    then reconsider y = x as Element of T;
    ex N being net of T st N is_eventually_in A & y is_a_cluster_point_of
    N by A2,Th21;
    hence thesis by A1;
  end;
  hence thesis;
end;
