(* s1 INCLUDED IN (OR OBS. EQUAL TO) s2 *)

Inductive incls(s1 s2:set):Prop:=
  exds2 : (forall z:dart, exds s1 z -> exds s2 z) 
          -> incls s1 s2.(* s1 INCLUDED IN (OR OBS. EQUAL TO) s2 *)

Inductive incls(s1 s2:set):Prop:=
  exds2 : (forall z:dart, exds s1 z -> exds s2 z) 
          -> incls s1 s2.

(* Specifications: *)
Lemma set_minus_s_Ds :forall(s1 s2:set)(x:dart), ~ exds s1 x -> exds s2 x -> set_minus s1 (Ds s2 x) = set_minus s1 s2.
Lemma card_minus_set:forall(s1 s2:set), incls s2 s1 -> (card (set_minus s1 s2) + card s2 = card s1)%nat.