theorem knockout_from_paper1 {ρ : ℝ} (hρ : 0 < ρ) {m : ℕ}
(hJ : 0 < J) (hm_pos : 0 < m) (hm_lt : m < J) :
0 < knockoutRetained J ρ m ∧ knockoutRetained J ρ m < 1 :=
knockout_partial_retained hρ hJ hm_pos hm_ltPropositions 12-17 and Corollary 5: