(* THEN, BY INDUCTION ON j: *)

Lemma bottom_B0_ter: forall(m:fmap)(z:dart)(i j:nat),
inv_hmap m -> exd m z -> ~pred m zero z ->
let zi := Iter (MA0.f m) i z in
let zj := Iter (MA0.f m) j z in
let m0 := B m zero zi in
(j <= i < MA0.Iter_upb m z)%nat ->
bottom m0 zero zj = bottom m zero zj.