
theorem Th33:
  for L being compact Hausdorff non empty TopSpace, N being net
  of L st for c, d being Point of L st c is_a_cluster_point_of N & d
  is_a_cluster_point_of N holds c = d holds for s being Point of L st s
  is_a_cluster_point_of N holds s in Lim N
proof
  let L be compact Hausdorff non empty TopSpace, N be net of L such that
A1: for c, d being Point of L st c is_a_cluster_point_of N & d
  is_a_cluster_point_of N holds c = d;
  let c be Point of L such that
A2: c is_a_cluster_point_of N;
  assume not c in Lim N;
  then consider M being subnet of N such that
A3: not ex P being subnet of M st c in Lim P by YELLOW_6:37;
  consider d being Point of L such that
A4: d is_a_cluster_point_of M by Th30;
A5: d is_a_cluster_point_of N by A4,Th31;
  c <> d by A3,A4,Th32;
  hence contradiction by A1,A2,A5;
end;
