
theorem
  for T being non empty TopStruct, S being sequence of T, x being Point
  of T st S is_convergent_to x holds x is_a_cluster_point_of S
proof
  let T be non empty TopStruct, S be sequence of T, x be Point of T;
  assume
A1: S is_convergent_to x;
  ex S1 being subsequence of S st S1 is_convergent_to x
  proof
    reconsider S1=S as subsequence of S by VALUED_0:19;
    take S1;
    thus thesis by A1;
  end;
  hence thesis by Th31;
end;
