A structured proof of the Kolmogorov superposition theorem

We present a well-structured detailed exposition of a well-known proof of the following celebrated result solving Hilberts 13th problem on superpositions. For functions of 2 variables the statement is as follows. Kolmogorov Theorem. There are continuous functions $varphi_1,ldots,varphi_5 : [,0, 1,]to [,0,1,]$ such that for any continuous function $f: [,0,1,]^2tomathbb R$ there is a continuous function $h: [,0,3,]tomathbb R$ such that for any $x,yin [,0, 1,]$ we have $$f(x,y)=sumlimits_{k=1}^5 hleft(varphi_k(x)+sqrt{2},varphi_k(y)right).$$ The proof is accessible to non-specialists, in particular, to students familiar with only basic properties of continuous functions.