+[1: apply (uparrow_to_in_segment s ⌊n,\fst (a \sub n)⌋ ? x H2);
+ simplify; intros; cases (a i); assumption;
+|2: intros;
+ lapply (uparrow_upperlocated a ≪x,h≫) as Ha1;
+ [2: apply (segment_preserves_uparrow s); assumption;]
+ lapply (segment_preserves_supremum s a ≪?,h≫ H2) as Ha2;
+ cases Ha2; clear Ha2;
+ cases (H1 a a); lapply (H5 H3 Ha1) as HaC;
+ lapply (segment_cauchy C s ? HaC) as Ha;
+ lapply (sigma_cauchy ? H ? x ? Ha); [left; assumption]
+ apply (restric_uniform_convergence C s ≪x,h≫ a Hletin)]
+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 hint_mah5:
+ ∀C. segment (hos_carr (os_r C)) → segment (hos_carr (os_l C)).
+ intros; assumption; qed.
+
+coercion hint_mah5 nocomposites.
+
+lemma hint_mah6:
+ ∀C. segment (hos_carr (os_l C)) → segment (hos_carr (os_r C)).
+ intros; assumption; qed.
+
+coercion hint_mah6 nocomposites.
+
+lemma restrict_uniform_convergence_downarrow:
+ ∀C:ordered_uniform_space.property_sigma C →
+ ∀s:segment (os_l C).exhaustive (segment_ordered_uniform_space C s) →
+ ∀a:sequence (segment_ordered_uniform_space C s).
+ ∀x:C. ⌊n,\fst (a n)⌋ ↓ x →
+ in_segment (os_l C) s x ∧ ∀h:x ∈ s.a uniform_converges ≪x,h≫.
+intros; split;
+[1: apply (downarrow_to_in_segment s ⌊n,\fst (a n)⌋ ? x); [2: apply H2];
+ simplify; intros; cases (a i); assumption;
+|2: intros;
+ lapply (downarrow_lowerlocated a ≪x,h≫) as Ha1;
+ [2: apply (segment_preserves_downarrow s a x h H2);]
+ lapply (segment_preserves_infimum s a ≪?,h≫ H2) as Ha2;
+ cases Ha2; clear Ha2;
+ cases (H1 a a); lapply (H6 H3 Ha1) as HaC;
+ lapply (segment_cauchy C s ? HaC) as Ha;
+ lapply (sigma_cauchy ? H ? x ? Ha); [right; assumption]
+ apply (restric_uniform_convergence C s ≪x,h≫ a Hletin)]
+qed.