
theorem Th24:
  for T being sober non empty TopSpace holds T is T_0
proof
  let T be sober non empty TopSpace;
  for p,q being Point of T st Cl{p} = Cl{q} holds p = q
  proof
    let p,q be Point of T such that
A1: Cl{p} = Cl{q};
    Cl{p} is irreducible by Th17;
    then consider r being Point of T such that
    r is_dense_point_of Cl{p} and
A2: for q being Point of T st q is_dense_point_of Cl{p} holds r = q by Def5;
    p = r by A2,Th18;
    hence thesis by A1,A2,Th18;
  end;
  hence thesis by Th23;
end;
