reserve a,b,c for set;
reserve r for Real,
  X for set,
  n for Element of NAT;

theorem
  for X being set holds ADTS X = {}-DiscreteTop(X)
proof
  let X be set;
  set T = {}-DiscreteTop(X);
A1: the carrier of T = X by Def8;
A2: cobool X = {{},X} by TEX_1:def 2;
A3: the topology of T c= cobool X
  proof
    let a be object;
    assume
A4: a in the topology of T;
    then reconsider a as Subset of T;
    a is open by A4;
    then a c= X & X is empty or a is non proper or a c= {} by A1,Th43;
    then a = {} or a = X by A1;
    hence thesis by A2,TARSKI:def 2;
  end;
  {}T = {};
  then
A5: {} in the topology of T by PRE_TOPC:def 2;
  [#]T = X by Def8;
  then X in the topology of T by PRE_TOPC:def 2;
  then cobool X c= the topology of T by A5,A2,ZFMISC_1:32;
  then the topology of T = cobool X by A3;
  hence thesis by A1,TEX_1:def 3;
end;
