enrich
   list-del
with
   functions 
       . ∪ . : list × list → list;
       . \ . : list × list → list;
      comment: x \ y : remove from x all elements that are in y;
       filter : list × list → list;
      comment: filter(x,y) : remove from x all elements that are not in y.;
   
   predicates 
       . ⊆ . : list × list;
   
   variables 
       x : list;
       y : list;
       z : list;
       a : elem;
       b : elem;
       c : elem;
       m : nat;
       n : nat;
   
   
   axioms 
      subset: x ⊆ y ↔ (∀ a. a ∈ x → a ∈ y);
      used for: ls;
      
      union: x ∪ y = rmdup(x + y);
      
      diff-e: [] \ y = [];
      used for: s, ls;
      
      diff-y: a ∈ y → (a ' + x) \ y = x \ y;
      used for: s, ls, lc;
      
      diff-n: ¬ a ∈ y → (a ' + x) \ y = a + (x \ y);
      used for: s, ls;
      
      filt-e: filter([], y) = [];
      used for: s, ls;
      
      filt-y: a ∈ y → filter(a ' + x, y) = a ' + filter(x, y);
      used for: s, ls, lc;
      
      filt-n: ¬ a ∈ y → filter(a ' + x, y) = filter(x, y);
      used for: s, ls;
   
   
end enrich