
theorem Th26:
  for T being non empty TopSpace holds T is Frechet implies T is sequential
proof
  let T be non empty TopSpace;
  assume
A1: T is Frechet;
  for A being Subset of T holds (for S being sequence of T st S is
  convergent & rng S c= A holds Lim S c= A) implies A is closed
  proof
    let A be Subset of T;
    assume
A2: for S being sequence of T st S is convergent & rng S c= A holds Lim S c= A;
A3: Cl(A) c= A
    proof
      let x be object;
      assume
A4:   x in Cl(A);
      then reconsider p=x as Point of T;
      consider S being sequence of T such that
A5:   rng S c= A and
A6:   p in Lim S by A1,A4;
      S is_convergent_to p by A6,Def5;
      then S is convergent;
      then Lim S c= A by A2,A5;
      hence thesis by A6;
    end;
    A c= Cl(A) by PRE_TOPC:18;
    then A = Cl(A) by A3;
    hence thesis by PRE_TOPC:22;
  end;
  hence thesis by Th25;
end;
