Proof of purchase doesn't have to be a retail receipt; a cheque stub, credit card or bank statement will also do. Having said that, the response you received was completely out of line since as you say the product is obviously still in warranty and you should raise a complaint (and make sure you CC your local Trading Standards office).

BTW, if you bought it with a credit card (or any other form of credit) the bank is jointly liable so you may want to contact them as well.