Title:A structured proof of Kolmogorov's Superposition Theorem

Abstract:We present a well-structured detailed exposition of a well-known proof of the following celebrated result solving Hilbert's 13th problem on superpositions. For functions of 2 variables the statement is as follows.
Kolmogorov Theorem. There are continuous functions φ1,…,φ5:[0,1]→[0,1] such that for any continuous function f:[0,1]2→R there is a continuous function h:[0,3]→R such that for any x,y∈[0,1] we have

f(x,y)=∑k=15h(φk(x)+2–√φk(y)).

The proof is accessible to non-specialists, in particular, to students familiar with only basic properties of continuous functions.