
theorem Th39:
  for T being non empty TopStruct, S being sequence of T, x being
  Point of T, n0 being Nat st x is_a_cluster_point_of S holds x
  is_a_cluster_point_of S^\n0
proof
  let T be non empty TopStruct, S be sequence of T, x be Point of T, n0 be Nat;
  assume
A1: x is_a_cluster_point_of S;
  set S1 = S^\n0;
  let O be Subset of T, n be Nat;
  assume O is open & x in O;
  then consider m0 being Element of NAT such that
A2: n + n0 <= m0 and
A3: S.m0 in O by A1;
  n0 in NAT & n0 <= n + n0 by NAT_1:11,ORDINAL1:def 12;
  then reconsider m=m0-n0 as Element of NAT by A2,INT_1:5,XXREAL_0:2;
  take m;
  thus n <= m by A2,XREAL_1:19;
  S1.m = S.(m0-n0+n0) by NAT_1:def 3
    .= S.m0;
  hence thesis by A3;
end;
