reserve Y for non empty TopStruct;

theorem
  for A being non empty Subset of Y holds A is anti-discrete & A is not
  trivial implies A is not T_0
proof
  let A be non empty Subset of Y;
  assume
A1: A is anti-discrete;
  consider s being object such that
A2: s in A by XBOOLE_0:def 1;
  reconsider s0 = s as Element of A by A2;
  assume A is not trivial;
  then A <> {s0};
  then consider t being object such that
A3: t in A and
A4: t <> s0 by ZFMISC_1:35;
  reconsider s, t as Point of Y by A2,A3;
  assume
A5: A is T_0;
  now
    per cases by A3,A4,A5;
    suppose
      ex E being Subset of Y st E is closed & s in E & not t in E;
      then consider E being Subset of Y such that
A6:   E is closed & s in E and
A7:   not t in E;
      A c= E by A1,A2,A6,TEX_4:def 2;
      hence contradiction by A3,A7;
    end;
    suppose
      ex F being Subset of Y st F is closed & not s in F & t in F;
      then consider F being Subset of Y such that
A8:   F is closed and
A9:   not s in F and
A10:  t in F;
      A c= F by A1,A3,A8,A10,TEX_4:def 2;
      hence contradiction by A2,A9;
    end;
  end;
  hence contradiction;
end;
