Please show the judgement by a real world court that emails sent by Game Protect are a written contract 
We can't, because you are not based in the real world !
If you are, in which jurisdiction are you based ? What is your company registration number ?
Also, please share the details to your lawyers, your sollicitor or both;
this is a pre-requisite to start the process of a real-world, legal lawsuit and you'll see very soon a judgement by a real world court !
If you are right in what you do, there is no problem and you will win in court, maybe even collect damages for defamation !
Ok, please quote the applicable law where it says that emails sent by Game Protect are a written contract?
Shit, so in your "real world" there are law texts regulating game protect's communication ?