Yes, you have to show proof of residency. Sometimes though, you can go to the school board and ask for transfer to another school, but there has to be a really good reason for that request. If it is a psychological reason, if other kids are bullying yours in the other school and damaging their confidence - you may be able to get a doctor or psychologist involved to have your child moved to another school. Please keep in mind, this is my personal view on this questions but may not be correct as the rules for that school board may not allow this - but all you can do is ask.
Good luck with everything!