
theorem Th40:
  for T being non empty TopStruct, S being sequence of T, x being
  Point of T st x is_a_cluster_point_of S holds x in Cl(rng S)
proof
  let T be non empty TopStruct, S be sequence of T, x be Point of T;
  assume
A1: x is_a_cluster_point_of S;
  for G being Subset of T st G is open holds x in G implies rng S meets G
  proof
    let G be Subset of T;
    assume
A2: G is open;
    assume x in G;
    then consider m being Element of NAT such that
    0 <= m and
A3: S.m in G by A1,A2;
    m in NAT;
    then m in dom S by NORMSP_1:12;
    then S.m in rng S by FUNCT_1:def 3;
    then S.m in rng S /\ G by A3,XBOOLE_0:def 4;
    hence thesis;
  end;
  hence thesis by PRE_TOPC:def 7;
end;
