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 Th21:
  for x, y being Point of Y holds y in MaxADSet(x) iff MaxADSet(y)
  = MaxADSet(x)
proof
  let x, y be Point of Y;
  MaxADSet(y) is maximal_anti-discrete by Th20;
  then
A1: MaxADSet(y) is anti-discrete;
A2: MaxADSet(x) is maximal_anti-discrete by Th20;
  then
A3: MaxADSet(x) is anti-discrete;
  thus y in MaxADSet(x) implies MaxADSet(y) = MaxADSet(x)
  proof
    assume y in MaxADSet(x);
    then
    MaxADSet(x) in {A where A is Subset of Y : A is anti-discrete & y in A
    } by A3;
    then MaxADSet(x) c= MaxADSet(y) by ZFMISC_1:74;
    hence thesis by A2,A1;
  end;
  assume
A4: MaxADSet(y) = MaxADSet(x);
  {y} c= MaxADSet(y) by Th12;
  hence thesis by A4,ZFMISC_1:31;
end;
