reserve X for non empty TopSpace,
  D for Subset of X;

theorem
  for Y being 1-element TopStruct st the topology of Y is non
  empty holds Y is almost_discrete implies Y is TopSpace-like
proof
  let Y be 1-element TopStruct;
  consider d being Element of Y such that
A1: the carrier of Y = {d} by Def1;
  reconsider D = {d} as Subset of Y;
  assume the topology of Y is non empty;
  then consider A being Subset of Y such that
A2: A in the topology of Y by SUBSET_1:4;
  assume
A3: for A being Subset of Y st A in the topology of Y holds (the carrier
  of Y) \ A in the topology of Y;
A4: bool D = {{},D} by ZFMISC_1:24;
  now
    per cases by A1,A4,TARSKI:def 2;
    suppose
A5:   A = {};
      D \ A in the topology of Y by A1,A2,A3;
      hence {{},D} c= the topology of Y by A2,A5,ZFMISC_1:32;
    end;
    suppose
A6:   A = D;
      D \ A in the topology of Y by A1,A2,A3;
      then {} in the topology of Y by A6,XBOOLE_1:37;
      hence {{},D} c= the topology of Y by A2,A6,ZFMISC_1:32;
    end;
  end;
  then the topology of Y = {{}, the carrier of Y} by A1,A4,XBOOLE_0:def 10;
  then reconsider Y as anti-discrete TopStruct by TDLAT_3:def 2;
  Y is TopSpace-like;
  hence thesis;
end;
