rw [← factorizes]
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)
g : (Quot fun a b => ElementaryMove R X a b) → FreeModuleAux R X := Quot.lift f wit
factorizes : ∀ (s : FormalSum R X), g (sum s) = sum ((a, x) :: s) := Quot.liftBeta f wit
⊢ sum ((a, x) :: s₁) = sum ((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)
g : (Quot fun a b => ElementaryMove R X a b) → FreeModuleAux R X := Quot.lift f wit
factorizes : ∀ (s : FormalSum R X), g (sum s) = sum ((a, x) :: s) := Quot.liftBeta f wit
⊢ g (sum s₁) = sum ((a, x) :: s₂)