rintro ⟨z, rfl, hz⟩
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
x y : M
this : x < y → (Exponential (x + 1) y ↔ ∃ z ≤ y, y = 2 * z ∧ Exponential x z)
hxy : x < y
⊢ (∃ z, y = 2 * z ∧ Exponential x z) → ∃ z ≤ y, y = 2 * z ∧ Exponential x z
case intro.intro
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
x z : M
hz : Exponential x z
this : x < 2 * z → (Exponential (x + 1) (2 * z) ↔ ∃ z_1 ≤ 2 * z, 2 * z = 2 * z_1 ∧ Exponential x z_1)
hxy : x < 2 * z
⊢ ∃ z_1 ≤ 2 * z, 2 * z = 2 * z_1 ∧ Exponential x z_1