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14
15 include "dama/models/nat_uniform.ma".  
16 include "dama/bishop_set_rewrite.ma".
17 include "dama/ordered_uniform.ma".
18
19 definition nat_ordered_uniform_space:ordered_uniform_space.
20  apply (mk_ordered_uniform_space (mk_ordered_uniform_space_ ℕ ℕ (refl_eq ? ℕ)));
21 simplify; intros 10;  cases H (_); cases (H7 y); apply H8; cases (H7 s);
22 lapply (H11 H1) as H13; apply (le_le_eq);
23 [2: apply (le_transitive ??? H5); apply (Le≪ ? H13); assumption;
24 |1: assumption;
25 |3: change with (le (os_r ℕ) (\snd y) (\fst y));
26     apply (ge_transitive ??? H5);apply (ge_transitive ???? H4);
27     change with (le (os_l ℕ) (\fst s) (\snd s));
28     apply (Le≫ ? H13); apply le_reflexive;
29 |4: assumption;]
30 qed. 
31  
32 interpretation "Ordered uniform space N" 'N = nat_ordered_uniform_space.