I'm reading this book:
Dung, N. V., Van Huynh, D., Smith, P. F., & Wisbauer, R. (1994). Extending modules (Vol. 313). CRC Press.
In the proof of Lemma 11.1, there is a part that I can not understand:
Let M be a module and $X \subset M$ be a direct summand and consider any exact sequence:
$0 \rightarrow K \rightarrow M \rightarrow X \rightarrow 0$.
My question is: If $N$ is a direct summand of $M$ that contains $K$, is the module $N/K$ isomorphic to a direct summand of $X$? The book states: Yes but it dos not explain why.
Here is the Lemma.
