
theorem Th31:
  for T being non empty TopStruct, S being sequence of T, x being
  Point of T st ex S1 being subsequence of S st S1 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;
  given S1 being subsequence of S such that
A1: S1 is_convergent_to x;
  let O be Subset of T, n be Nat;
  assume O is open & x in O;
  then consider n1 being Nat such that
A2: for m being Nat st n1 <= m holds S1.m in O by A1;
  reconsider n2=max(n1,n) as Element of NAT by ORDINAL1:def 12;
A3: S1.n2 in O by A2,XXREAL_0:25;
  consider NS being increasing sequence of NAT such that
A4: S1 = S * NS by VALUED_0:def 17;
  take NS.n2;
  n <= n2 & n2 <= NS.n2 by SEQM_3:14,XXREAL_0:25;
  hence n <= NS.n2 by XXREAL_0:2;
  n2 in NAT;
  then n2 in dom NS by FUNCT_2:def 1;
  hence thesis by A4,A3,FUNCT_1:13;
end;
