You do not have to have the receipt, you need proof of purchase from them. For example, a bank or credit card statement showing the transaction. A receipt just makes it much easier to prove that you bought it there.
If you cannot provide any such proof, they are well within their rights. How do they know you did not get it elsewhere?
Remember you don't have to have a receipt although it obviously helps. You have the same rights even if you lose your receipt. If you want to return faulty goods and you don't have the receipt, you may need to show 'proof of purchase'. A credit or bank statement should do, or even the word of a shop assistant who sold the item to you.