
theorem Th34:
  for T being non empty TopStruct, S being sequence of T, x,y
  being Point of T st for n being Element of NAT holds S.n = y & S
  is_convergent_to x holds x in Cl({y})
proof
  let T be non empty TopStruct, S be sequence of T, x,y be Point of T;
  assume
A1: for n being Element of NAT holds S.n = y & S is_convergent_to x;
  for G being Subset of T st G is open holds x in G implies {y} meets G
  proof
    let G be Subset of T;
    assume
A2: G is open;
    assume x in G;
    then consider n being Nat such that
A3: for m being Nat st n <= m holds S.m in G by A1,A2,FRECHET:def 3;
A4: n in NAT by ORDINAL1:def 12;
    S.n in G by A3;
    then
A5: y in G by A1,A4;
    y in {y} by TARSKI:def 1;
    then y in {y} /\ G by A5,XBOOLE_0:def 4;
    hence thesis;
  end;
  hence thesis by PRE_TOPC:def 7;
end;
