∀ {R : Type u_1} {S : Type u_2} [inst : NonUnitalNonAssocSemiring R] [inst_1 : NonUnitalNonAssocSemiring S]
  (f : R →ₙ+* S) (g : S → R) (h₁ : Function.LeftInverse g ⇑f) (h₂ : Function.RightInverse g ⇑f) (x y : S),
  (↑((↑f).inverse g h₁ h₂)).toFun (x + y) = (↑((↑f).inverse g h₁ h₂)).toFun x + (↑((↑f).inverse g h₁ h₂)).toFun y