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That depends upon what you mean when you say "Must I...?"
If you are asking if it will 'work' when you try to install and use that Vista Home Premium then yes, it will. It'll install just fine and, if it has been a reasonable while since you last installed and activated it on the original PC, it'll activate and validate just fine too. It'll 'work' in a practical sense.
But if, by asking "Must I...?" you meant "Will it give me a legal install?" then no. It won't. It'll just give you a working install. The only way you can get another 'legal' install is by purchasing another license. Unless you have a volume licensing scheme arrangement, a Windows license can only legally/legitimately be used on one PC at a time.