+ 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]