proposition: Lemma ne_total_lt `{!TrivialApart A} x y : x <> y -> x < y |_| y < x.
proof: Proof. intros neq;apply trivial_apart in neq. apply apart_total_lt. assumption. Qed.
imports: HoTT.Classes.interfaces.abstract_algebra HoTT.Classes.interfaces.orders HoTT.Classes.theory.apartness