∀ {M : Type u_3} [inst : Monoid M] {A : Type u_1} [inst_1 : AddMonoid A] [inst_2 : DistribMulAction M A] {B₁ : Type u_2}
  [inst_3 : AddMonoid B₁] [inst_4 : DistribMulAction M B₁] (f : A →+[M] B₁) (g : B₁ → A)
  (h₁ : Function.LeftInverse g ⇑f) (h₂ : Function.RightInverse g ⇑f) (x y : B₁),
  (↑((↑f).inverse g h₁ h₂)).toFun (x + y) = (↑((↑f).inverse g h₁ h₂)).toFun x + (↑((↑f).inverse g h₁ h₂)).toFun y