If the rule statement serves as the thesis sentence for a longer discussion about a legal rule that has developed over time in a series of cases, the rule proof serves as your explanation and elaboration of that thesis sentence. In the rule proof you can support the rule statement by reference to a statute or to judicial holdings and opinions, further explain key terms or phrases you have used in the rule statement, or define the scope of the rule by more detailed reference to judicial opinions, including the holdings and the factual contexts in which they arise. In this section, you discuss case facts, holdings, and reasoning only to support and explicate the rule statement.
As you are writing the rule proof section, remember the reader is best served (i) when you emphasize those case facts or extracts of the opinions that support or explain your rule statement; and (ii) when you make an explicit statement about the relationship of the case to your rule; in fact, this is an excellent way to introduce a case because it tells the reader right at the outset why you are bringing up that case at all.
In other words, you do not want your readers to be thinking, "Why am I reading this?" as they read about the cases. Do not make them wonder about the relevance of what they are reading, even for a short time, or make them do the work of inferring the relevance themselves. And, as you move from paragraph to paragraph in a longer rule proof, use thesis sentences to connect important ideas in the rule statement with an elaboration of these ideas in the rule proof. Each paragraph should develop a single concept (thesis); successive paragraphs should have a demonstrable relationship to that concept - providing an additional illustration of it, extending it, contrasting it, or moving to a related but different category of idea.