" If not, you can't write that they both begin then, unless you are OK with leaving that up in the air, maybe because if they miss the dinner, they can't get in to see the program. " You can't write "Dinner and program begins" because the reader sees no immediate justification for lumping the two things together in a single conceptual unit, even though they might actually be one, and hears a grammar mistake before he can catch up. The "remove one subject" rule is for the case of pronouns, not for number.
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