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This is a good question. Let me first give the answer and then give the rationale.
"The idea that everyone would laugh at him and me and that after we would not mind being laughed at again."
The reason why "him" and "me" are correct is twofold.
First, "he" and "I" cannot be the correct answer, because "he" and "I" are subject pronouns. In other words, we would only use "he" and "I" if they were the subject of the sentence or if you had a predicate nominative. So, by the process of elimination, you could arrive at the correct answer.
Second, the preposition "at" governs "him" and "me." In other words, the preposition does not take a subject pronoun but an object pronoun.
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