-notation "'segment_preserves_supremum'" non associative with precedence 90 for @{'segment_preserves_supremum}.
-notation "'segment_preserves_infimum'" non associative with precedence 90 for @{'segment_preserves_infimum}.
-
-interpretation "segment_preserves_supremum" 'segment_preserves_supremum = (h_segment_preserves_supremum (os_l _)).
-interpretation "segment_preserves_infimum" 'segment_preserves_infimum = (h_segment_preserves_supremum (os_r _)).
-
+lemma segment_preserves_infimum:
+ ∀O:ordered_set.∀l,u:O.∀a:sequence {[l,u]}.∀x:{[l,u]}.
+ ⌊n,\fst (a n)⌋ is_decreasing ∧
+ (\fst x) is_infimum ⌊n,\fst (a n)⌋ → a ↓ x.
+intros; split; cases H; clear H;
+[1: apply H1;
+|2: cases H2; split; clear H2;
+ [1: apply H;
+ |2: clear H; intro y0; apply (H3 (\fst y0));]]
+qed.
+