The point I believe you may be missing, Appolinaria, is that one cannot prove universal nonexistence. It's an impossible request, and is used merely to evade the burden of proof which is implicitly held by the one making a positive claim. It's really nothing but hand-waving... smoke and mirrors... a "hey, look over there!" while the person runs away...
Do they leave behind any evidence of their existence? No.
But there is evidence that's unaccounted for. Like our finite universe and why it began. So it's a free for all until we can answer a few questions.
P.S.- It's entertaining to watch when atheists all team up to battle the crazy theists, not realizing they're doing exactly what they claim to be against.
Further, it's not about "ganging up" or "battling theists," but instead about holding their theological claims to the same standards to which we hold all other claims. Claims of the theist deserve no special deference, and frankly the fact they don't receive unearned deference or respect does not ipso facto mean people are "doing exactly what they claim to be against."
Let's please try to maintain some perspective here. You're basically pointing and laughing and going "hahaha... tee hee hee... silly atheists..." when in fact you appear to be operating under a misapprehension.
proving non-existence: when an arguer cannot provide the evidence for his claims, he may challenge his opponent to prove it doesn't exist (e.g., prove God doesn't exist; prove UFO's haven't visited earth, etc.). Although one may prove non-existence in special limitations, such as showing that a box does not contain certain items, one cannot prove universal or absolute non-existence, or non-existence out of ignorance. One cannot prove something that does not exist. The proof of existence must come from those who make the claims.