Generally speaking, you are required to complete a "proof of loss" form in which you itemize your items, along with an estimate of fair market value (or replacement value). In most of these losses (e.g. fire loss), the "insured" does not have receipts for their items, and it is unreasonable for the insurer to require a receipt for all of the items. Photographs of the items would be useful, but those are not necessary either. The insurer can take your "examination under oath" also to explore these losses.
As for the value of the items, you can also find comparable items on the Internet (or Wal Mart of whatever) and provide them with that list, to determine value (but this of course does not substantiate that you had the item). At the "end of the day," you may just have your word on these losses.
You do have to cooperate with the insurer, but you are not required to provide a receipt. You may want to keep working with the adjuster and demand payment for the loss. If they don't pay, then you may have to look for a lawyer to represent you. If the insurer has been very unreasonable, then you may even have a claim for "bad faith."
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I have looked into the California case law and other authorities, and I cannot find anything directly on this point. The general authorities I have found indicate that you simply have to offer the same type of proof that you would offer in court, which could be your testimony if you have nothing else.
You probably want to make sure that you have a copy of the policy (the entire policy, not just the "declarations" page e.g.) and see what it says about how you must prove your claim. If the adjuster insists on having a receipt, then you may want to ask him or her to explain to you why they are requesting this, especially for such old items. They do have a duty to deal with you in good faith. It is usually best to not start with threats about your recovering punitive damages from them, but if you have a valid claim, then ultimately they should be concerned about such a claim.
If you were to pursue the claim in court, your sworn testimony would be sufficient to assert your claim. Of course, the finder-of-fact (e.g. jury) could accept or reject your testimony.
If the policy allows the insurer the right to take your "examination under oath," then I generally think it's a good idea to offer to allow them to take such an examination if they have questions about the loss; it tends to make the insurer look more unreasonable if they refuse your offer.
I am sorry that I don't have anything more specific to offer for you to use with the adjuster. Please let me know if you have any more questions.
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