reserve X for non empty TopSpace;

theorem Th2:
  for x being Point of X holds
  Cl {x} = meet {F where F is Subset of X : F is closed & x in F}
proof
  let x be Point of X;
  set G = {F where F is Subset of X : F is closed & x in F};
  set H = {F where F is Subset of X : F is closed & {x} c= F};
  now
    let P be object;
    assume P in G;
    then consider F being Subset of X such that
A1: F = P and
A2: F is closed and
A3: x in F;
    {x} c= F by A3,ZFMISC_1:31;
    hence P in H by A1,A2;
  end;
  then
A4: G c= H;
  now
    let P be object;
    assume P in H;
    then consider F being Subset of X such that
A5: F = P and
A6: F is closed and
A7: {x} c= F;
    x in F by A7,ZFMISC_1:31;
    hence P in G by A5,A6;
  end;
  then
A8: H c= G;
  Cl {x} = meet H by Th1;
  hence thesis by A4,A8,XBOOLE_0:def 10;
end;
