-lemma segment_preserves_supremum:
- ∀O:ordered_set.∀l,u:O.∀a:sequence {[l,u]}.∀x:{[l,u]}.
- ⌊n,\fst (a n)⌋ is_increasing ∧
- (\fst x) is_supremum ⌊n,\fst (a n)⌋ → a ↑ x.
+
+lemma x2sx:
+ ∀O:half_ordered_set.
+ ∀s:segment O.∀x,y:half_segment_ordered_set ? s.
+ \fst x ≰≰ \fst y → x ≰≰ y.
+intros 4; cases x; cases y; clear x y; simplify; unfold hos_excess;
+whd in ⊢ (?→? (% ? ?) ? ? ? ?); simplify in ⊢ (?→%);
+cases (wloss_prop O) (E E); do 2 rewrite < E; intros; assumption;
+qed.
+
+lemma sx2x:
+ ∀O:half_ordered_set.
+ ∀s:segment O.∀x,y:half_segment_ordered_set ? s.
+ x ≰≰ y → \fst x ≰≰ \fst y.
+intros 4; cases x; cases y; clear x y; simplify; unfold hos_excess;
+whd in ⊢ (? (% ? ?) ? ? ? ? → ?); simplify in ⊢ (% → ?);
+cases (wloss_prop O) (E E); do 2 rewrite < E; intros; assumption;
+qed.
+
+lemma h_segment_preserves_supremum:
+ ∀O:half_ordered_set.∀s:segment O.
+ ∀a:sequence (half_segment_ordered_set ? s).
+ ∀x:half_segment_ordered_set ? s.
+ increasing ? ⌊n,\fst (a n)⌋ ∧
+ supremum ? ⌊n,\fst (a n)⌋ (\fst x) → uparrow ? a x.