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 Th20:
  for x being Point of Y holds MaxADSet(x) is maximal_anti-discrete
proof
  let x be Point of Y;
A1: for D being Subset of Y st D is anti-discrete & MaxADSet(x) c= D holds
  MaxADSet(x) = D
  proof
    let D be Subset of Y;
    assume
A2: D is anti-discrete;
    assume
A3: MaxADSet(x) c= D;
    {x} c= MaxADSet(x) by Th12;
    then {x} c= D by A3;
    then x in D by ZFMISC_1:31;
    then D c= MaxADSet(x) by A2,Th19;
    hence thesis by A3;
  end;
  MaxADSet(x) is anti-discrete by Th13;
  hence thesis by A1;
end;
