
theorem Th4:
  for T being discrete non empty TopSpace holds T is T_2
proof
  let T be discrete non empty TopSpace;
  let p, q be Point of T such that
A1: not p = q;
  take W = {p}, V = {q};
  thus W is open & V is open by TDLAT_3:15;
  W /\ V c= {}
  proof
    let a be object;
    assume
A2: a in W /\ V;
    then a in W by XBOOLE_0:def 4;
    then
A3: a = p by TARSKI:def 1;
    assume not a in {};
    a in V by A2,XBOOLE_0:def 4;
    hence contradiction by A1,A3,TARSKI:def 1;
  end;
  hence p in W & q in V & W /\ V = {} by TARSKI:def 1;
end;
