have : ∃ v ∈ a ^ˢ k, ∃ u ∈ a, s = v ⁀' u := by simpa [mem_seqProduct_iff] using hs
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* 𝐈𝚺₁
a k : M
ih : ∀ {s : M}, s ∈ a ^ˢ k ↔ Seq s ∧ lh s = k ∧ ∀ (i z : M), ⟪i, z⟫ ∈ s → z ∈ a
s : M
hs : s ∈ a ^ˢ k ×ˢ a
⊢ Seq s ∧ lh s = k + 1 ∧ ∀ (i z : M), ⟪i, z⟫ ∈ s → z ∈ a
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* 𝐈𝚺₁
a k : M
ih : ∀ {s : M}, s ∈ a ^ˢ k ↔ Seq s ∧ lh s = k ∧ ∀ (i z : M), ⟪i, z⟫ ∈ s → z ∈ a
s : M
hs : s ∈ a ^ˢ k ×ˢ a
this : ∃ v ∈ a ^ˢ k, ∃ u ∈ a, s = v ⁀' u
⊢ Seq s ∧ lh s = k + 1 ∧ ∀ (i z : M), ⟪i, z⟫ ∈ s → z ∈ a