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

theorem
  for x, y being Point of X holds Cl {x} c= Cl {y} iff MaxADSet(y) c=
  meet {G where G is Subset of X : G is open & x in G}
proof
  let x, y be Point of X;
  set FY = {G where G is Subset of X : G is open & y in G};
  FY c= bool the carrier of X
  proof
    let C be object;
    assume C in FY;
    then ex P being Subset of X st C = P & P is open & y in P;
    hence thesis;
  end;
  then reconsider FY as Subset-Family of X;
  set FX = {G where G is Subset of X : G is open & x in G};
  FX c= bool the carrier of X
  proof
    let C be object;
    assume C in FX;
    then ex P being Subset of X st C = P & P is open & x in P;
    hence thesis;
  end;
  then reconsider FX as Subset-Family of X;
  thus Cl {x} c= Cl {y} implies MaxADSet(y) c= meet {G where G is Subset of X
  : G is open & x in G}
  proof
    assume Cl {x} c= Cl {y};
    then
A1: meet FY c= meet FX by Th51;
    (Cl {y}) /\ meet FY c= meet FY by XBOOLE_1:17;
    hence thesis by A1;
  end;
  {y} c= MaxADSet(y) by Th12;
  then
A2: y in MaxADSet(y) by ZFMISC_1:31;
  assume MaxADSet(y) c= meet {G where G is Subset of X : G is open & x in G};
  then
A3: y in meet FX by A2;
  set G = (Cl {y})`;
  assume
A4: not Cl {x} c= Cl {y};
  not x in Cl {y} by A4,TOPS_1:5,ZFMISC_1:31;
  then x in G by SUBSET_1:29;
  then G in FX;
  then
A5: meet FX c= G by SETFAM_1:3;
  {y} c= Cl {y} by PRE_TOPC:18;
  then y in Cl {y} by ZFMISC_1:31;
  hence contradiction by A5,A3,XBOOLE_0:def 5;
end;
