(*======================================================== EXISTENCY PROPERTIES OF ITERATED DARTS, upb IN ORBITS...: WITH NOETHERIAN IND. =======================================================*)
(* FOR THE FOLLOWING LEMMA: *)
Definition P1
(m:fmap)(z:dart)(s:set):Prop:=
let sr := Iter_rem_aux m z s in
let n := Iter_upb_aux m z s in
~ exds sr (Iter (f m) n z).
(* THE upb_ITERATED DART BY f DOES NOT REMAIN: *)

Lemma not_exds_rem_upb :
forall(m:fmap)(z:dart)(s:set),
let sr := Iter_rem_aux m z s in
let n := Iter_upb_aux m z s in
~ exds sr (Iter (f m) n z).