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 A being Subset of X holds A is anti-discrete & A is closed iff for
  x being Point of X st x in A holds A = Cl {x}
proof
  let A be Subset of X;
  thus A is anti-discrete & A is closed implies for x being Point of X st x in
  A holds A = Cl {x}
  by ZFMISC_1:31,TOPS_1:5;
  thus (for x being Point of X st x in A holds A = Cl {x}) implies A is
  anti-discrete & A is closed
  proof
    assume
A1: for x being Point of X st x in A holds A = Cl {x};
    then for x be Point of X st x in A holds A c= Cl {x};
    hence A is anti-discrete;
    hereby
      per cases;
      suppose
        A is empty;
        hence thesis;
      end;
      suppose
        A is non empty;
        then consider a being object such that
A2:     a in A;
        reconsider a as Point of X by A2;
        A = Cl {a} by A1,A2;
        hence thesis;
      end;
    end;
  end;
end;
