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 Th17:
  for A being non empty Subset of Y holds A is anti-discrete & A
  is closed implies A is maximal_anti-discrete
proof
  let A be non empty Subset of Y;
  assume
A1: A is anti-discrete;
  assume
A2: A is closed;
  for D being Subset of Y st D is anti-discrete & A c= D holds A = D
  proof
    let D be Subset of Y;
    assume D is anti-discrete;
    then D misses A or D c= A by A2;
    then
A3: D /\ A = {} or D c= A;
    assume A c= D;
    hence thesis by A3,XBOOLE_1:28;
  end;
  hence thesis by A1;
end;
