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
  for Y being non empty TopSpace,
      x being Point of Y holds
    MaxADSet(x) c= meet {F where F is Subset of Y : F is closed & x in F}
proof
  let Y be non empty TopSpace;
  let x be Point of Y;
  set G = {F where F is Subset of Y : F is closed & x in F};
  [#]Y in G; then
A1: G <> {};
  G c= bool the carrier of Y
  proof
    let C be object;
    assume C in G;
    then ex P being Subset of Y st C = P & P is closed & x in P;
    hence thesis;
  end;
  then reconsider G as Subset-Family of Y;
  now
    let C be set;
    assume C in G;
    then ex F being Subset of Y st F = C & F is closed & x in F;
    hence MaxADSet(x) c= C by Th23;
  end;
  hence thesis by SETFAM_1:5,A1;
end;
