+ ∀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: unfold in H2; cases H2; clear H2;unfold in H3 H4; cases H4; clear H4; unfold in H2;
+ cases (wloss_prop (os_l C)) (W W); apply prove_in_segment; unfold; rewrite <W;
+ simplify;
+ [ apply (le_transitive ?? x ? (H2 O));
+ lapply (under_wloss_upperbound (os_r C) ?? (h_segment_upperbound (os_r C) s a) O) as K;
+ unfold in K; whd in K:(?%????); simplify in K; rewrite <W in K; apply K;
+ | intro; cases (H5 ? H4); clear H5 H4;
+ lapply (under_wloss_upperbound (os_l C) ?? (segment_upperbound s a) w) as K;
+ unfold in K; whd in K:(?%????); simplify in K; rewrite <W in K;
+ apply K; apply H6;
+ | intro; unfold in H4; rewrite <W in H4;
+ lapply depth=0 (H5 (seg_u_ (os_l C) s)) as k; unfold in k:(%???→?);
+ simplify in k; rewrite <W in k; lapply (k
+ simplify;intro; cases (H5 ? H4); clear H5 H4;
+ lapply (under_wloss_upperbound (os_l C) ?? (segment_upperbound s a) w) as K;
+ unfold in K; whd in K:(?%????); simplify in K; rewrite <W in K;
+ apply K; apply H6;
+
+
+
+ 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);