The argument would go like this:
Take as a given that Everything is lost.
Ok then, answer the question. If the premise is true then what isn't lost.
The answer is nothing.
Therefore saying that Everything is lost is the opposite of saying there is nothing that has not been lost.
If you premise that nothing is lost then answer the following question:
What is not lost? The answer is everything. Saying that everything is not lost is logically EXACTLY the same thing as saying nothing is lost which is not what you meant to say.
If you premise that SOME things are lost, answer this question:
What is lost? The answer is SOME things but not everything. So if you want to say that some things have been lost but not everything, you should say not all is lost, because that is what means that not all is lost.