You haven't proven anything until you've proven it to a court. As with your other posts, hire an attorney an proceed cautiously. The terms of the contract control and certain notices and communications should likely be made. Have an attorney review thw contract and the facts and write it for you. It could be win, lose, or nobody wins based on what you do now. Spend the money on the ounce of prevention rather than the pound of cure. Hire an attorney before you do anything else.
This response does not create an attorney-client relationship and is not intended to provide legal advice for your specific situation.
I would send them written notice of the breach of contract and failure to pay. You may want to engage an attorney to send them a letter. The small cost of engaging counsel and having an attorney send a letter often results in a payment or settlement attempt, as the same sends a message that you mean business.