+
+definition lt ≝ λE:ordered_set.λa,b:E. a ≤ b ∧ a # b.
+
+interpretation "ordered sets less than" 'lt a b = (lt _ a b).
+
+lemma lt_coreflexive: ∀E.coreflexive ? (lt E).
+intros 2 (E x); intro H; cases H (_ ABS);
+apply (bs_coreflexive ? x ABS);
+qed.
+
+lemma lt_transitive: ∀E.transitive ? (lt E).
+intros (E); unfold; intros (x y z H1 H2); cases H1 (Lxy Axy); cases H2 (Lyz Ayz);
+split; [apply (le_transitive ???? Lxy Lyz)] clear H1 H2;
+cases Axy (H1 H1); cases Ayz (H2 H2); [1:cases (Lxy H1)|3:cases (Lyz H2)]clear Axy Ayz;
+[1: cases (os_cotransitive ??? y H1) (X X); [cases (Lxy X)|cases (os_coreflexive ?? X)]
+|2: cases (os_cotransitive ??? x H2) (X X); [right;assumption|cases (Lxy X)]]
+qed.
+
+theorem lt_to_excess: ∀E:ordered_set.∀a,b:E. (a < b) → (b ≰ a).
+intros (E a b Lab); cases Lab (LEab Aab); cases Aab (H H);[cases (LEab H)]
+assumption;
+qed.