theorem fragility_bounds {ρ : ℝ} (hρ : 0 < ρ) (hJ : 0 < J) (hJ2 : 1 < J) :
0 < fragilityIndex J ρ ∧ fragilityIndex J ρ < 1 := by
simp only [fragilityIndex]
have hk := knockout_partial_retained hρ hJ (by omega : 0 < 1) (by omega : 1 < J)
constructor
· linarith [hk.2]
· linarith [hk.1]Propositions 12-17 and Corollary 5: