
theorem Th15:
  for T being non empty TopStruct, S being sequence of T, S1 being
  subsequence of S, x being Point of T holds S is_convergent_to x implies S1
  is_convergent_to x
proof
  let T be non empty TopStruct, S be sequence of T, S1 be subsequence of S, x
  be Point of T;
  assume
A1: S is_convergent_to x;
  let U1 be Subset of T;
  assume U1 is open & x in U1;
  then consider n being Nat such that
A2: for m being Nat st n <= m holds S.m in U1 by A1;
  take n;
  let m be Nat;
  assume
A3: n <= m;
  m in NAT by ORDINAL1:def 12;
  then
A4: m in dom S1 by NORMSP_1:12;
  consider NS being increasing sequence of NAT such that
A5: S1 = S * NS by VALUED_0:def 17;
  m <= NS.m by SEQM_3:14;
  then S.(NS.m) in U1 by A2,A3,XXREAL_0:2;
  hence S1.m in U1 by A5,A4,FUNCT_1:12;
end;
