There are actually two issues here. The first with the customer and the second with Discovery.
As to the customer, you have a minor problem in that you can't really bind a customer to a warranty that they didn't have access to prior to the purchase. However, you can make the argument that they could have returned the machine at that point and you would have given them a full refund.
An "express warranty" is one that is written out and explained in detail. An "implied warranty" is one that is based on a reasonable person standard and just means that a warranty is what a reasonable person would think it was under the same circumstances or what is normal in the industry.
Their lawyer will likely argue that instead, the failure to provide a warranty prior to the purchase means that there is no "express warranty" and therefore it is based on a "reasonable" standard. I don't know what your specific judge would do in a case like that but your lawyer can likely tell you since they will know that specific judge. Most judges that I've been in front of , and that's a lot, would likely say that it just meant they could have returned the photo booth at that time and you would have had to give them a full refund. However, you also want to look to see if any materials you send them, your website description (f you have one), etc. has info on the warranty because if it does then they can no longer argue they didn't have access to it.
As to the Discover issue, what they can and can't do will depend on your agreement with them or your credit card processor. Most of those say that the credit card company reserves the right to be the ultimate arbitrator (decision maker) in disputes so they could, in that kind of contract, give them customer a 100% refund. You could still pursue the customer since a credit card company issuing a refund doesn't mean that you were in the wrong legally, it just means they decided they wanted to give a refund. You need to get a copy of your agreement with them and then let a local lawyer review it as well.
At this point the best course of action is to get all of your paperwork together and let the lawyer write a letter to Discover and to the customer telling them your position, explaining the warranty, explaining the law on "express warrranties" and then explaining what you will sue for if it becomes necessary.
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