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I am sorry to hear about this situation. The fact that there was only an oral agreement to the work does not invalidate the work. The oral agreement is still a binding contract. The estimate is proof of the agreement, as was the work itself.
The customer took advantage of the credit card's policy to challenge the charge. The credit card company will only reverse the charge if the customer can show that the charge was unauthorized, which he likely lied and stated that it was.
At this point, someone in your situation has two options:
1) File a LIEN. See HERE for an explanation of this is done. This may force them to pay, or the lien would not be released; or
2) File a small claims court matter for breach of contract and money had and received. To sue in a state court, one needs to have a "cause of action." There are numerous causes of action, such as "breach of contract," "negligence," "fraud," "unjust enrichment," etc., as well as causes of action rooted in statutory law. Every state has their own although they are very similar to each other in every state because they all stem from the same common law. A pleading in Court needs at least one cause of action, although it is not unusual to have more than one.
The elements of a breach of contract claim are: (1) the existence of a valid contract; (2) the plaintiff's performance or tendered performance; (3) the defendant's breach of the contract; and (4) damages as a result of the breach. Marketshare Telecom, L.L.C. v. Ericsson, Inc., 198 S.W.3d 908, 923 (Tex. Ct. App. 2006).
Money had and received is a cause of action where money is held by a party that rightfully belongs to you.
This would be a small claims court case. See HERE. Although the directions are for Harris County, they are very good and give a general overview for any Texas county.
A letter threatening this may be enough to have them pay without having to file anything.
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