This contract clause [...] that I have this-and-that right.
which of the following can I use instead of the [...]?
- something else?
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I would not use mandate to describe something you want to do. A contract is usually written, so it doesn't technically say anything. Dictate usually means one of the preceding options. Stipulate is technically the act of making the agreement, so it'd be more accurate to say something like "Didn't we stipulate that I have x right." if you use that word. The contract can show, prove or indicate that you have the right.
As long as we're being finicky, although the right can be used in this sense, it often implies a higher authority. Therefore I suggest use the word privilege rather than right in this case, since a contract is an instrument used to demonstrate mutual agreement and your rights may under certain legal conditions be revoked.
"The contract shows/indicates/demonstrates/proves that I have this-and-that privilege."
Which of those words you choose is primarily a matter of opinion. I like indicate the best, since I think it is an elegant and firm but not too firm. Proves has more strength and show is a bit more colloquial. Demonstrate sounds a little verbose for this application to me and I would prefer to use it prior to closing a deal since its abbreviation, demo (Merriam-Webster Online), is often used specifically to show off products, like computer software Demo. versions. These opinions are not firm but I offer them anyway so that we do not just end up back exactly where we started.
With the exception of demo. as shown above, I'm primarily referencing the definitions from Noah Webster's 1828 American Dictionary of the English Language. My links include the Webster's Revised Unabridged 1913 definitions too. I probably should reference a legal dictionary in this case but critically speaking, Legalese is an entirely different language from English with lots of common-law baggage and is hence difficult to accurately research and may be considered off-topic. Few, if any of us here are legal experts. I certainly am not.