Linear Transformation of Generated Module/Proof 2

Proof
This proof assumes that $R$ is a ring with unity, so $G$ and $H$ become unitary modules.

Let $y \in G$ be arbitrary.

Then by definition of generator, $y$ is the linear combination of elements of $S$:


 * $\ds y = \sum_{k \mathop = 1}^n \lambda_k a_k$

for $a_1, a_2, \ldots, a_n \in S, \lambda_1, \lambda_2, \ldots, \lambda n \in R$

Then: