Help? Client is demanding a full refund?

Hey There

I have worked with loads of people for graphic and web design projects, for my safety and theirs I always make us both sign a contract that outlines things like delivery, cancellation policy and what happens as far money goes if something goes pear shaped.

I have had a client who paid me a deposit of $80 last week for a web design and development project - I started the project and sent over a preliminary mockup to him, he instantly didn't like it and without reason or understanding that it was no where near finished - he started demanding a full refund. I sent him a polite offer back saying its a difficult situation because 9 hours had already gone into the project. By giving a full refund, this guy has just wasted a full days work that could have been spent on other clients and projects. He was a polite guy so I mailed him saying I'm prepared to offer a 50% refund due to the fact many hours have already been spent on the design.

Basically he's said that's totally unacceptable and he is now threatening to contact PayPal and take it further.
I've never actually had a dodgy client until now - I should have picked up on this a little earlier because he has a company based in the US yet he paid me on paypal from his sisters account? Which I found a little strange but put a blind eye to it and kept going onwards.

He's digitally signed a contract that states the following for cancellation

Cancellation. Both parties understand that Client or Designer may terminate the service at any time if, for any reason, the relationship is deemed unsatisfactory by either party. Upon written or verbal cancellation, Client is responsible for payment for all expenses incurred and any work done toward the completion of the project based on the percentage of project completed. Should Client cancel the project following its completion, Client is responsible for full payment as per the above estimate plus all other expenses incurred.

I just want to check that I'm definitely safe with this?

Thanks
Matt