
theorem Th13:
  for T being non empty TopStruct, A being Subset of T, S being
  sequence of T, x being Point of T st rng S c= A & x in Lim S holds x in Cl(A)
proof
  let T be non empty TopStruct, A be Subset of T, S be sequence of T, x be
  Point of T;
  assume that
A1: rng S c= A and
A2: x in Lim S;
  for O being Subset of T st O is open holds x in O implies A meets O
  proof
    let O be Subset of T;
    assume
A3: O is open;
A4: S is_convergent_to x by A2,FRECHET:def 5;
    assume x in O;
    then consider n being Nat such that
A5: for m being Nat st n <= m holds S.m in O by A3,A4;
    n in NAT by ORDINAL1:def 12;
    then n in dom S by NORMSP_1:12;
    then
A6: S.n in rng S by FUNCT_1:def 3;
    S.n in O by A5;
    then S.n in A /\ O by A1,A6,XBOOLE_0:def 4;
    hence thesis;
  end;
  hence thesis by PRE_TOPC:def 7;
end;
