The image reads:
If $T_n$ is exact, that is to say, tensoring with $N$ transforms all exact sequences into exact sequences, then $N$ is said to be a flat $A$-module.
Proposition 2.19: The following are equivalent for any $A$ module $N$: (i) $N$ is flat (ii) If $0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$ is any exact sequence of $A$ modules, the tensored sequence $0 \rightarrow M' \otimes N \rightarrow M \otimes N \rightarrow M'' \otimes N \rightarrow 0$ is exact
As far as I understand, (i) and (ii) are identical: (i) says $N$ is flat, (ii) spills out the definition of flat. What is there to prove? The book says:
(i) is implied and implied by (ii) by taking a long exact sequence and splitting into short exact sequences
I don't understand what long exact sequence I need to consider, because to me, the proposition looks like a tautology.