+
+ cases H2 (Ha Hx); clear H2; cases Hx; split;
+ lapply depth=0 (under_wloss_upperbound (os_l C) ?? (segment_upperbound s a) O) as W1;
+ lapply depth=0 (under_wloss_upperbound (os_r C) ?? (h_segment_upperbound (os_r C) s a) O) as W2;
+ lapply (H2 O); simplify in Hletin; simplify in W2 W1;
+ cases a in Hletin W2 W1; simplify; cases (f O); simplify; intros;
+ whd in H6:(? % ? ? ? ?);
+ simplify in H6:(%);
+ cases (wloss_prop (os_l C)); rewrite <H8 in H5 H6 ⊢ %;
+ [ change in H6 with (le (os_l C) (seg_l_ (os_l C) s) w);
+ apply (le_transitive ??? H6 H7);
+ | apply (le_transitive (seg_u_ (os_l C) s) w x H6 H7);
+ |
+ lapply depth=0 (supremum_is_upper_bound ? x Hx (seg_u_ (os_l C) s)) as K;
+ lapply depth=0 (under_wloss_upperbound (os_l C) ?? (segment_upperbound s a));
+ apply K; intro; lapply (Hletin n); unfold; unfold in Hletin1;
+ rewrite < H8 in Hletin1; intro; apply Hletin1; clear Hletin1; apply H9;
+ | lapply depth=0 (h_supremum_is_upper_bound (os_r C) ⌊n,\fst (a n)⌋ x Hx (seg_l_ (os_r C) s)) as K;
+ lapply depth=0 (under_wloss_upperbound (os_r C) ?? (h_segment_upperbound (os_r C) s a));
+ apply K; intro; lapply (Hletin n); unfold; unfold in Hletin1;
+whd in Hletin1:(? % ? ? ? ?);
+simplify in Hletin1:(%);
+ rewrite < H8 in Hletin1; intro; apply Hletin1; clear Hletin1; apply H9;
+
+
+ apply (segment_upperbound ? l);
+ generalize in match (H2 O); generalize in match Hx; unfold supremum;
+ unfold upper_bound; whd in ⊢ (?→%→?); rewrite < H4;
+ split; unfold; rewrite < H4; simplify;
+ [1: lapply (infimum_is_lower_bound ? ? Hx u);
+
+
+
+split;
+ [1: apply (supremum_is_upper_bound ? x Hx u);
+ apply (segment_upperbound ? l);
+ |2: apply (le_transitive l ? x ? (H2 O));
+ apply (segment_lowerbound ? l u a 0);]
+|2: intros;
+ lapply (uparrow_upperlocated a ≪x,h≫) as Ha1;
+ [2: apply (segment_preserves_uparrow C l u);split; assumption;]
+ lapply (segment_preserves_supremum C l u a ≪?,h≫) as Ha2;
+ [2:split; assumption]; cases Ha2; clear Ha2;
+ cases (H1 a a); lapply (H6 H4 Ha1) as HaC;
+ lapply (segment_cauchy ? l u ? HaC) as Ha;
+ lapply (sigma_cauchy ? H ? x ? Ha); [left; split; assumption]
+ apply restric_uniform_convergence; assumption;]
+qed.
+
+lemma hint_mah1:
+ ∀C. Type_OF_ordered_uniform_space1 C → hos_carr (os_r C).
+ intros; assumption; qed.
+
+coercion hint_mah1 nocomposites.
+
+lemma hint_mah2:
+ ∀C. sequence (hos_carr (os_l C)) → sequence (hos_carr (os_r C)).
+ intros; assumption; qed.
+
+coercion hint_mah2 nocomposites.
+
+lemma hint_mah3:
+ ∀C. Type_OF_ordered_uniform_space C → hos_carr (os_r C).
+ intros; assumption; qed.
+
+coercion hint_mah3 nocomposites.
+
+lemma hint_mah4:
+ ∀C. sequence (hos_carr (os_r C)) → sequence (hos_carr (os_l C)).
+ intros; assumption; qed.
+
+coercion hint_mah4 nocomposites.
+
+lemma restrict_uniform_convergence_downarrow:
+ ∀C:ordered_uniform_space.property_sigma C →
+ ∀l,u:C.exhaustive {[l,u]} →
+ ∀a:sequence {[l,u]}.∀x: C. ⌊n,\fst (a n)⌋ ↓ x →
+ x∈[l,u] ∧ ∀h:x ∈ [l,u].a uniform_converges ≪x,h≫.
+intros; cases H2 (Ha Hx); clear H2; cases Hx; split;
+[1: split;
+ [2: apply (infimum_is_lower_bound ? x Hx l);
+ apply (segment_lowerbound ? l u);
+ |1: lapply (ge_transitive ? ? x ? (H2 O)); [apply u||assumption]
+ apply (segment_upperbound ? l u a 0);]
+|2: intros;
+ lapply (downarrow_lowerlocated a ≪x,h≫) as Ha1;
+ [2: apply (segment_preserves_downarrow ? l u);split; assumption;]
+ lapply (segment_preserves_infimum C l u a ≪x,h≫) as Ha2;
+ [2:split; assumption]; cases Ha2; clear Ha2;
+ cases (H1 a a); lapply (H7 H4 Ha1) as HaC;
+ lapply (segment_cauchy ? l u ? HaC) as Ha;
+ lapply (sigma_cauchy ? H ? x ? Ha); [right; split; assumption]
+ apply restric_uniform_convergence; assumption;]
+qed.