I'm studying module-theory on my own and in the chapter I'm currently in, the book discusses tensor product of modules (actually I decided to read this book only because of this chapter xD), there's a theorem in the book that talks about the tensor product of a given exact sequence (the book doesn't say it this way though, but this is how I memorize the whole theorem because It's kinda like that). I've attached the theorem and its proof, the book that I'm reading is not in English so I was forced to translate everything into English, if there are points that are ambiguous tell me to clarify them.
I understand the whole proof, except two parts that I've highlighted them in red in the pdf. Please do not tell me how I could prove the theorem on my own, I want someone to clarify the claims of the book, I'm sure there might be many other ways to prove it, but for now I only want to understand the parts that I've highlighted them in red. I know what I should prove and I understand the strategy completely but I fail to understand those two claims, so please just explain the reason behind those two claims.
Thanks in advance (By the way, it's my first post on this forum. Hi people out there! )