proposition: Lemma not_lt_apart_lt_flip x y : ~(x < y) -> x ≶ y -> y < x.
proof: Proof. intros nlt neq. apply apart_iff_total_lt in neq. destruct neq. - destruct nlt;auto. - auto. Qed.
imports: HoTT.Classes.interfaces.abstract_algebra HoTT.Classes.interfaces.orders HoTT.Classes.theory.apartness