Zabreiko's Lemma with Bicomplex and hyperbolic scalars and its applications

In this paper, we shall consider the notion of hyperbolic semi norm which on a module $X$ to set of all positive hyperbolic numbers. We shall prove the characterization of continuity of hyperbolic semi norm in this setup. We shall prove Zabreiko's lemma when $X$ is a F, $\mathbb{BC}$ module, where $\mathbb{BC}$ denotes the set of Bi complex numbers.(analogous to completeness). This lemma shall be used to prove the fundamental theorems of functional analysis like the Closed Graph Theorem, Open mapping Theorem, Uniform Boundedness principle.