reserve X for non empty TopSpace;
reserve Y for non empty TopStruct;
reserve x for Point of Y;
reserve Y for non empty TopStruct;

theorem Th39:
  for Y being non empty TopSpace,
      A being Subset of Y holds
    MaxADSet(A) c= meet {G where G is Subset of Y : G is open & A c= G}
proof
  let Y be non empty TopSpace;
  let A be Subset of Y;
  set F = {G where G is Subset of Y : G is open & A c= G};
  [#]Y in F; then
A1: F <> {};
  F c= bool the carrier of Y
  proof
    let C be object;
    assume C in F;
    then ex P being Subset of Y st C = P & P is open & A c= P;
    hence thesis;
  end;
  then reconsider F as Subset-Family of Y;
  now
    let C be set;
    assume C in F;
    then ex G being Subset of Y st G = C & G is open & A c= G;
    hence MaxADSet(A) c= C by Th38;
  end;
  hence thesis by A1,SETFAM_1:5;
end;
