let wit : (s₁ s₂ : FormalSum R X) → (ElementaryMove R X s₁ s₂) → f s₁ = f s₂ := by
  intro s₁ s₂ hyp
  apply Quot.sound
  apply ElementaryMove.cons
  assumption
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s₁ s₂ : FormalSum R X
a : R
x : X
h : s₁ ≃ s₂
f : FormalSum R X → FreeModuleAux R X := fun s => sum ((a, x) :: s)
⊢ (a, x) :: s₁ ≃ (a, x) :: s₂
R : Type
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type
inst✝ : DecidableEq X
s₁ s₂ : FormalSum R X
a : R
x : X
h : s₁ ≃ s₂
f : FormalSum R X → FreeModuleAux R X := fun s => sum ((a, x) :: s)
wit : ∀ (s₁ s₂ : FormalSum R X), ElementaryMove R X s₁ s₂ → f s₁ = f s₂ :=
  fun s₁ s₂ hyp => Quot.sound (ElementaryMove.cons a x s₁ s₂ hyp)
⊢ (a, x) :: s₁ ≃ (a, x) :: s₂