An analog of the Hille theorem for hypercomplex functions in a finite-dimensional commutative algebra

We prove that a locally bounded and differentiable in the sense of Gâteaux function given in a finite-dimensional commutative Banach algebra over the complex field is also differentiable in the sense of Lorc and it is a monogenic function. The algebra Aₙᵐ has the Cartan basis for which the first m basic vectors I₁, I₂, …, Iₘ are idempotents, and next $n-m$ basis vectors Iₘ₊₁,Iₘ₊₂,,Iₙ are nilpotent elements.Every locally bounded and differentiable in the sense of Gâteaux function Φ Ωₙᵐ can be represented in the form of linear combination of these idempotents, nilpotents and corresponding Cauchy-type integrals.