(* 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.