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The removal of your unit before making measurements for the new unit was negligent and I am sure the other installer who installed the proper system told you that and would testify to that in court. Under IL law, you can bring a suit in small claims court for anything up to $10,000, and you need no attorney to go to small claims court.
You could likely get assistance with preparation from court though the local legal aid office, especially if you tell them you do not expect them to actually represent you.
You have grounds for a negligence case against them to seek the damages for the 10 days you were inconvenienced by their negligence plus the additional $3000 you had to pay over the price they gave you. The total inconvenience damages would generally be about equal to the additional cost you incurred over the original contract, but you should sue for the whole $10,000 and negotiate down to the $6000 ($3000 extra costs plus $3000 inconvenience) with them.
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