
theorem Th16:
  for T being non empty TopStruct, S being sequence of T, S1 being
  subsequence of S holds Lim S c= Lim S1
proof
  let T be non empty TopStruct, S be sequence of T, S1 be subsequence of S;
  let x be object;
  assume
A1: x in Lim S;
  then reconsider x9=x as Point of T;
  S is_convergent_to x9 by A1,FRECHET:def 5;
  then S1 is_convergent_to x9 by Th15;
  hence thesis by FRECHET:def 5;
end;
