∀ {R : Type u} {A : Type v} {B : Type w} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Semiring B]
  [inst_3 : Algebra R A] [inst_4 : Algebra R B] (toEquiv toEquiv_1 : A ≃ B) (e_toEquiv : toEquiv = toEquiv_1)
  (map_mul' : ∀ (x y : A), toEquiv.toFun (x * y) = toEquiv.toFun x * toEquiv.toFun y)
  (map_add' : ∀ (x y : A), toEquiv.toFun (x + y) = toEquiv.toFun x + toEquiv.toFun y)
  (commutes' : ∀ (r : R), toEquiv.toFun ((algebraMap R A) r) = (algebraMap R B) r),
  { toEquiv := toEquiv, map_mul' := map_mul', map_add' := map_add', commutes' := commutes' } =
    { toEquiv := toEquiv_1, map_mul' := ⋯, map_add' := ⋯, commutes' := ⋯ }