(* *)
(**************************************************************************)
-include "models/nat_dedekind_sigma_complete.ma".
+include "models/increasing_supremum_stabilizes.ma".
include "models/nat_ordered_uniform.ma".
lemma nat_us_is_oc: ∀s:‡ℕ.order_continuity (segment_ordered_uniform_space ℕ s).
intros 3; split; intros 3; cases H1; cases H2; clear H2 H1; cases H; clear H;
-[1: cases (nat_dedekind_sigma_complete s a H1 ? H2);
+[1: cases (increasing_supremum_stabilizes s a H1 ? H2);
exists [apply w1]; intros;
apply (restrict nat_ordered_uniform_space s ??? H4);
generalize in match (H ? H5);
generalize in match (a i) in ⊢ (?? % ? → %); intro X; cases X; clear X;
intro; rewrite < H9 in H7; simplify; rewrite < H7; simplify;
apply (us_phi1 nat_uniform_space ? H3); simplify; apply (eq_reflexive (us_carr nat_uniform_space));
-|2: cases (nat_dedekind_sigma_complete_r s a H1 ? H2);
+|2: cases (increasing_supremum_stabilizes_r s a H1 ? H2);
exists [apply w1]; intros;
apply (restrict nat_ordered_uniform_space s ??? H4);
generalize in match (H ? H5);