suffices ∃ t, Seq t ∧ lh t = lh s ∧ ∀ i < lh s, ∀ x < s, ⟪i, x⟫ ∈ s → ⟪i, f x⟫ ∈ t
by  rcases this with ⟨t, Ht, hts, h⟩
    exact ⟨t, Ht, hts, fun i x hx ↦ h i (Hs.lt_lh_of_mem hx) x (lt_of_mem_rng hx) hx⟩
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
f : M → M
hf : 𝚺₁-Function₁ f
s : M
Hs : Seq s
⊢ ∃ t, Seq t ∧ lh t = lh s ∧ ∀ (i x : M), ⟪i, x⟫ ∈ s → ⟪i, f x⟫ ∈ t
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
f : M → M
hf : 𝚺₁-Function₁ f
s : M
Hs : Seq s
⊢ ∃ t, Seq t ∧ lh t = lh s ∧ ∀ i < lh s, ∀ x < s, ⟪i, x⟫ ∈ s → ⟪i, f x⟫ ∈ t