have : 2 * ((a / i) / 2) + 1 = a / i := by simpa [LenBit.iff_rem.mp h] using div_add_mod (a / i) 2
case mp
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
i a : M
hi : Pow2 i
h : LenBit i a
⊢ ∃ k, ∃ r < i, a = k * (2 * i) + i + r
case mp
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
i a : M
hi : Pow2 i
h : LenBit i a
this : 2 * (a / i / 2) + 1 = a / i
⊢ ∃ k, ∃ r < i, a = k * (2 * i) + i + r