Direct consequence of a Lemma used in definition of Quotient Module

Question

I have the following Lemma already proved.

If $A$ is a submodule of the $R$-module $X$, then $\{r(x+a) | a \in A \} \subset rx+A$.

Then in the same book it is said that as a direct consequence of this Lemma, the coset $rx+A$ depends only of $r \in R$ and the coset $x+A$, and it is used to define the multiplication $\otimes :R \times X/A \rightarrow X/A$. But I cannot see how this Lemma is a direct consequence of this statement, Its possible to show this with more steps?

Answer 1

What needs to be shown is that if $x+A=y+A$ then $rx+A=ry+A$. Now, if $x+A=y+A$ then $y-x\in A$, so $y=x+a$ for some $a\in A$. So, by your lemma, $ry=r(x+a)\in rx+A$. So, the cosets $rx+A$ and $ry+A$ are equal.