rcases show ‖S‖ ≤ i * ‖L‖ ∨ i * ‖L‖ < ‖S‖ from le_or_lt _ _ with (le | lt)
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚫₀ + 𝛀₁
z S L i : M
⊢ z = S{L}[i] ↔ (‖S‖ ≤ i * ‖L‖ → z = 0) ∧ (i * ‖L‖ < ‖S‖ → ∃ b ≤ S, Exponential (i * ‖L‖) b ∧ z = S / b % L # 1)
case inl
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚫₀ + 𝛀₁
z S L i : M
le : ‖S‖ ≤ i * ‖L‖
⊢ z = S{L}[i] ↔ (‖S‖ ≤ i * ‖L‖ → z = 0) ∧ (i * ‖L‖ < ‖S‖ → ∃ b ≤ S, Exponential (i * ‖L‖) b ∧ z = S / b % L # 1)

case inr
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚫₀ + 𝛀₁
z S L i : M
lt : i * ‖L‖ < ‖S‖
⊢ z = S{L}[i] ↔ (‖S‖ ≤ i * ‖L‖ → z = 0) ∧ (i * ‖L‖ < ‖S‖ → ∃ b ≤ S, Exponential (i * ‖L‖) b ∧ z = S / b % L # 1)