Here are two details in the proofs I can't understand in this chapter.
1) On page 108, at the begining of the proof of Proposition 10.13, Atiyah claims that "$a$-adic completion commutes with finite direct sums" and he says this fact follows from (10.3):
Take for example $G'=G_1$, $G''=G_2$ and $G$ be their direct sum, then how can I see that the second exact squence is split exact? I find some functorial explanations about this on MSE but I haven't learned homological algebra so it maybe answered by an elementary method.
2) On page 112, I feel confused about the last three lines of the proof of Lemma 10.23. How does this imply ii) of the Lemma?
And this is 10.2:
