theorem euler_identity_at_uniform (hJ : 0 < J) (ε₀ : ℝ) :
∑ _j : Fin J, ε₀ * ((1 : ℝ) / ↑J) = ε₀ := by
rw [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]
have hJne : (↑J : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (by omega)
field_simpTheorems 5-7, Corollaries 2-4, Propositions 8-11: