Algebraic operations on analytic function germs

Let $F_{1}$ and $F_{2}$ be two analytic functions in a domain $G$. Prove that the functions $F_{1}+F_{2}$ and $F_{1} \cdot F_{2}$, defined by elements of the form $(B_{r}(a), f_{1}+f_{2})$, $\left(B_{r}(a), f_{1} \cdot f_{2}\right)$, where $\left(B_{r}(a), f_{1}\right)$ and $\left(B_{r}(a), f_{2}\right)$ are any elements of $F_{1}$ and $F_{2}$, are also analytic functions in $G$.

Let $F_{1}$ and $F_{2}$ be two analytic functions in a domain $G$, defined by elements of the form $(B_{r}(a), f_{1})$ and $(B_{r}(a), f_{2})$, where for every element $\left(B_{r}(a), f_{2}\right)$ of the function $F_{2}$ the regular function $f_{2}(z) \neq 0, z \in B_{r}(a)$. Prove that the function $\frac{F_{1}}{F_{2}}$, defined by elements of the form $\left(B_{r}(a), \frac{f_{1}}{f_{2}}\right)$, is analytic in the domain $G$.

Let $F$ be an analytic function in a domain $G$, defined by elements of the form $(B_{r}(a), f)$. Let $g: \mathbb {C} \rightarrow \mathbb {C}$ be a regular function on the plane $\mathbb {C}$. Prove that the function defined by elements of the form $\left(B_{r}(a), g(f)\right)$ is analytic in the domain $G$.

Let the initial element $(B_{r}(a), f_{0})$ generate an analytic function in a domain $G \subset \mathbb {C}$. Prove that in the domain $G$ there exists an analytic function with initial element