Canonical Injection into Cartesian Product of Modules/Proof 1

Proof
To demonstrate that $\inj_j$ is an epimorphism, we need to show that:


 * $(1): \quad \inj_j$ is an injection
 * $(2): \quad \forall x, y \in G_j: \map {\inj_j} {x +_j y} = \map {\inj_j} x + \map {\inj_j} y$
 * $(3): \quad \forall x_j \in G_j: \forall \lambda \in R: \map {\inj_j} {\lambda \circ_j x_j} = \lambda \circ \map {\inj_j} {x_j}$

Criteria $(1)$ and $(2)$ are a direct application of Canonical Injection is Monomorphism.

Let $x_j \in G_j$ be arbitrary.

Then we have: