reserve Y for non empty TopStruct;

theorem Th9:
  for A being Subset of Y holds A is discrete implies A is T_0
proof
  let A be Subset of Y;
  assume
A1: A is discrete;
  now
    let x, y be Point of Y;
    assume that
A2: x in A and
A3: y in A;
    {x} c= A by A2,ZFMISC_1:31;
    then consider G being Subset of Y such that
A4: G is open and
A5: A /\ G = {x} by A1,TEX_2:def 3;
    assume
A6: x <> y;
    now
      take G;
      thus G is open by A4;
      x in {x} by TARSKI:def 1;
      hence x in G by A5,XBOOLE_0:def 4;
      now
        assume y in G;
        then y in {x} by A3,A5,XBOOLE_0:def 4;
        hence contradiction by A6,TARSKI:def 1;
      end;
      hence not y in G;
    end;
    hence (ex V being Subset of Y st V is open & x in V & not y in V) or ex W
    being Subset of Y st W is open & not x in W & y in W;
  end;
  hence thesis;
end;
