Collapsible proof of the existence of the n-th root

This is a Czech prototype of a proof reducible to the very basics. I would like for a great database of mathematical proofs to exist, where all theorems would be proven in this level of detail, i.e. by expanding sufficiently, even a high school student could understand the most complex proofs.

Also, ever sice I created this page, this proof became my personal favourite proof in mathematical analysis.