The following theorem is from the Abstract Algebra by Dummit and Foote (in the section 10.4 tensor products of modules):
[![enter image description here](https://i.stack.imgur.com/wTRL2.png)](https://i.stack.imgur.com/wTRL2.png)

Would anybody illustrate how Theorem 8 is used to get
$$
\textrm{ker }\iota\subset\textrm{ker }\varphi
$$
in the proof? 

---

[![enter image description here](https://i.stack.imgur.com/G3bpm.png)](https://i.stack.imgur.com/G3bpm.png)