In this part of the paper (see, \textit{I. P. Melnichenko} and \textit{S. A. Plaksa} [Ukr. Mat. Zh. 48, 1518-1529 (1996); ibid. 1695-1703 (1996)] for the first and the second parts) the authors consider algebras of monogenic and \({\mathbf C}\)-analytic functions over a complex Banach algebra \( H_{\mathbf C}=H\oplus iH,\) where \(H\) is a Banach algebra obtained from the Banach space \(l_{1}\) of infinite sequences of real numbers by means of introducing of an associative and commutative operation. A number of results which generalized statements of the classical theory of analytic functions (criterion of monogeneity of functions, integral Cauchy formulas, Runge approximation theorems, analogy of symmetry principle, uniqueness theorems for \({\mathbf C}\)-analytic functions) are proved. Finally, a solution to the system \[ y \frac{\partial \varphi}{\partial x}=\frac{\partial \psi}{\partial y},\quad y \frac{\partial \varphi}{\partial y}=-\frac{\partial \psi}{\partial x}, \quad \psi (x,0) \equiv 0. \] is given.