∀ {S : Type u_2} {R : Type u_1} {c₁ c₂ c₃ : R} [inst : Semiring S] [inst_1 : AddCommGroup R] [inst_2 : Module S R]
  (x : S) (x_1 : QuaternionAlgebra R c₁ c₂ c₃),
  (QuaternionAlgebra.addEquivProd c₁ c₂ c₃).toAddMonoidHom (x • x_1) =
    (QuaternionAlgebra.addEquivProd c₁ c₂ c₃).toAddMonoidHom (x • x_1)