People who are saying that the contract isn't legal, the contract IS considered legit. The paper binds the agreement because it serves as proof of the transaction. This is not a billion dollar transaction and even if they brought this to the police, the paper is enough to serve as proof. Even if it's not the police, they could bring it to the yakuza (if they have ties with them) so it still works either way.
Remember at the beginning of this arc, Takeshi had alot of trouble getting job? He COULD beat the people up, but he understood he can't keep living as a deadbeat and being a famous delinquent in the end is just a title, so he tried to keep everything straight.
Yeah, time fuck them up then call the animal service to seize the place, Takeshi.