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;

theorem
  for x being Point of X, D being Subset of X st D is anti-discrete & Cl
  {x} c= D holds D = Cl {x}
proof
  let x be Point of X, D be Subset of X;
  assume D is anti-discrete;
  then D misses Cl {x} or D c= Cl {x};
  then
A1: D /\ Cl {x} = {} or D c= Cl {x};
  assume Cl {x} c= D;
  hence thesis by A1,XBOOLE_1:28;
end;
