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Registration of a business and name is XXXXX XXXXX of rights to the name. This is a REBUTTABLE presumption and you can prove your rights by proving that you were the first to use the name in commerce and seek to stop the other party from using the name. Proving that you were first to use in commerce can be done by showing that you were operating and conducting business through your records, such as sales receipts, sale tax records etc.
You need to start by sending them a cease and desist letter informing them that their using the same name causes a likelihood of confusion and give them the date you were first to use the name in the state. If they refuse to change their name, then you will need to go to court to sue over the rights to use the name and you will need an attorney at least at that point to pursue them to seek your rights to the name.
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