(* EQUALITY/DISJONCTION OF SETS *)
Definition impls(s1 s2:set):Prop:=
forall z:dart, exds s1 z -> exds s2 z.
Definition eqs(s1 s2:set):Prop:=
forall z:dart, exds s1 z <-> exds s2 z.
Definition disjs(s1 s2:set):Prop:=
forall z:dart, exds s1 z -> exds s2 z -> False.
(*=========================================================== NOETHERIAN RELATION ON (MULTI-)SETS ===========================================================*)
Definition Rs(sy sx:set):=
(card sy < card sx)%nat.

Lemma exds_Rs_Ds:
forall(s:set)(z:dart),
exds s z -> Rs (Ds s z) s.