
theorem Th3:
  for T being discrete non empty TopSpace holds T is regular
proof
  let T be discrete non empty TopSpace;
  let p be Point of T, P be Subset of T such that
  P <> {} and
  P is closed and
A1: p in P`;
  take W = {p}, V = P;
  thus W is open & V is open by TDLAT_3:15;
A2: not p in P by A1,XBOOLE_0:def 5;
  W /\ V c= {}
  proof
    let a be object;
    assume
A3: a in W /\ V;
    assume not a in {};
    a in W & a in V by A3,XBOOLE_0:def 4;
    hence contradiction by A2,TARSKI:def 1;
  end;
  hence p in W & P c= V & W /\ V = {} by TARSKI:def 1;
end;
