Proofs are like characters because the proofs have to be well devolved and convincing and so do characters! Good characters are well- rounded : they learn, change, and grow. Just like how when reading your proofs the persons opinions might change.
Transitions are like the plot in a story. The plot is a chain of events that happen in the story. Transitions move from one point to the next. (One event to the next event)
A paragraph is like a book because lots of the parts in a book are in a paragraph. They are in the story and paragraph for similar reasons. And the parts add to the story/paragraph.