A structured proof of Kolmogorov's Superposition Theorem

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 varphi1,ldots,varphi5 : [,0, 1,]→ [,0,1,] such that for any continuous function f: [,0,1,]2→mathbb R there is a continuous function h: [,0,3,]→mathbb R such that for any x,y∈ [,0, 1,] we have f(x,y)=sumlimitsk=15 hleft(varphik(x)+√2,varphik(y)right). The proof is accessible to non-specialists, in particular, to students familiar with only basic properties of continuous functions.