The following is from Diamond's Modular Forms book:
I don't understand the first three lines after the lemma, at all. Also, from which part of the lemma it is a consequence? I don't see a connection between the three mentioned lines ("One consequence of the lemma ...") and no part of the lemma!
Simple clear explanation would be much appreciated.
This is mostly a consequence of part $c$. In short, if you write a matrix $\gamma$ in terms of the two generating matrices $T, S$, perhaps as $\gamma = S T S$, then weak modularity for $S$ and $T$ implies weak modularity for $\gamma$.
You can see this by noting that $$ [\gamma]_k = [S]_k [T]_k [S]_k,$$ so that $$ [\gamma]_k f = [S]_k [T]_k ([S]_k f) = [S]_k ([T]_k f) = [S]_k f = f,$$ where for each of the three right equalities we use that $f$ is invariant under the slash operator for the two generating matrices.