+lemma h_segment_upperbound:
+ ∀C:half_ordered_set.
+ ∀s:segment C.
+ ∀a:sequence (half_segment_ordered_set C s).
+ upper_bound ? ⌊n,\fst (a n)⌋ (seg_u C s).
+intros 4; simplify; cases (a n); simplify; unfold in H;
+cases (wloss_prop C); rewrite < H1 in H; simplify; cases H;
+assumption;
+qed.
+
+notation "'segment_upperbound'" non associative with precedence 90 for @{'segment_upperbound}.
+notation "'segment_lowerbound'" non associative with precedence 90 for @{'segment_lowerbound}.
+
+interpretation "segment_upperbound" 'segment_upperbound = (h_segment_upperbound (os_l _)).
+interpretation "segment_lowerbound" 'segment_lowerbound = (h_segment_upperbound (os_r _)).
+
+lemma h_segment_preserves_uparrow:
+ ∀C:half_ordered_set.∀s:segment C.∀a:sequence (half_segment_ordered_set C s).
+ ∀x,h. uparrow C ⌊n,\fst (a n)⌋ x → uparrow (half_segment_ordered_set C s) a ≪x,h≫.
+intros; cases H (Ha Hx); split;
+[ intro n; intro H; apply (Ha n); apply (sx2x ???? H);
+| cases Hx; split;
+ [ intro n; intro H; apply (H1 n);apply (sx2x ???? H);
+ | intros; cases (H2 (\fst y)); [2: apply (sx2x ???? H3);]
+ exists [apply w] apply (x2sx ?? (a w) y H4);]]
+qed.
+
+notation "'segment_preserves_uparrow'" non associative with precedence 90 for @{'segment_preserves_uparrow}.
+notation "'segment_preserves_downarrow'" non associative with precedence 90 for @{'segment_preserves_downarrow}.
+
+interpretation "segment_preserves_uparrow" 'segment_preserves_uparrow = (h_segment_preserves_uparrow (os_l _)).
+interpretation "segment_preserves_downarrow" 'segment_preserves_downarrow = (h_segment_preserves_uparrow (os_r _)).
+
+(* Fact 2.18 *)
+lemma segment_cauchy:
+ ∀C:ordered_uniform_space.∀s:‡C.∀a:sequence {[s]}.
+ a is_cauchy → ⌊n,\fst (a n)⌋ is_cauchy.
+intros 6;
+alias symbol "pi1" (instance 3) = "pair pi1".
+alias symbol "pi2" = "pair pi2".
+apply (H (λx:{[s]} squareB.U 〈\fst (\fst x),\fst (\snd x)〉));
+(unfold segment_ordered_uniform_space; simplify);
+exists [apply U] split; [assumption;]
+intro; cases b; intros; simplify; split; intros; assumption;
+qed.
+