reserve X for non empty TopSpace,
  D for Subset of X;
reserve D for non empty set,
  d0 for Element of D;

theorem
  STS(D,d0) = ADTS(D) iff D = {d0}
proof
  set G = {P where P is Subset of D : d0 in P & P <> D};
  thus STS(D,d0) = ADTS(D) implies D = {d0}
  proof
    set d1 = the Element of D \ {d0};
    assume
A1: STS(D,d0) = ADTS(D);
    assume
A2: D <> {d0};
A3: now
      assume D \ {d0} = {};
      then D c= {d0} by XBOOLE_1:37;
      hence contradiction by A2,XBOOLE_0:def 10;
    end;
    then reconsider d1 as Element of D by XBOOLE_0:def 5;
    reconsider P = {d1} as Subset of D;
A4: d0 <> d1 by A3,ZFMISC_1:56;
    then not ex Q being Subset of D st Q = P & d0 in Q & Q <> D by TARSKI:def 1
;
    then not P in G;
    then
A5: P in {{},D} by A1,XBOOLE_0:def 5;
    {d0} c= D;
    then {d0} c= {d1} by A5,TARSKI:def 2;
    hence contradiction by A4,ZFMISC_1:18;
  end;
  assume
A6: D = {d0};
  then G = {} by Lm2;
  hence thesis by A6,ZFMISC_1:24;
end;
