induction s₂ with
| nil => 
  simp [mulMonom, coords]
| cons head tail ih =>
  simp [mulMonom, coords, mul]
  rw [← append_coords]
  rw [← append_coords]
  rw [ih]
  simp [add_assoc]
R : Type
inst✝³ : Ring R
inst✝² : DecidableEq R
G : Type
inst✝¹ : Group G
inst✝ : DecidableEq G
x₀ : G
s₁ s₂ s₃ : FormalSum R G
⊢ coords (mul s₁ (s₂ ++ s₃)) x₀ = coords (mul s₁ s₂) x₀ + coords (mul s₁ s₃) x₀
no goals