An Analogue of Laurent's Theorem for a Simply Connected Region

Our purpose is to establish the following theorem.

Theorem I. Given open disks D₁ and D₂ in the complexplane such that D = D₁ ∩ D₂ is non-void, and a function f on D with values in a (non-commutative) Banach algebra over the complex field such that f is holomorphic and reciprocable on D, then f is the product of two functions f₁ and f₂ such that f₁ is holomorphic and reciprocable on D₁ and f₂ is holomorphic and reciprocable on D₂.