reserve X for non empty TopSpace;

theorem Th1:
  for A being Subset of X holds
  Cl A = meet {F where F is Subset of X: F is closed & A c= F}
proof
  let A be Subset of X;
  set G = {F where F is Subset of X : F is closed & A c= F};
A1: G c= bool the carrier of X
  proof
    let C be object;
    assume C in G;
    then ex P being Subset of X st C = P & P is closed & A c= P;
    hence thesis;
  end;
  [#]X in G;
  then reconsider G as non empty Subset-Family of X by A1;
  now
    let P be set;
    assume P in G;
    then ex F being Subset of X st F = P & F is closed & A c= F;
    hence A c= P;
  end;
  then
A2: A c= meet G by SETFAM_1:5;
  A c= Cl A by PRE_TOPC:18;
  then Cl A in G;
  then
A3: meet G c= Cl A by SETFAM_1:3;
  now
    let S be Subset of X;
    assume S in G;
    then ex F being Subset of X st F = S & F is closed & A c= F;
    hence S is closed;
  end;
  then G is closed by TOPS_2:def 2;
  then Cl A c= meet G by A2,TOPS_1:5,TOPS_2:22;
  hence thesis by A3;
end;
