An estimate of the dimension of the null spaces of linear superpositions

In this article it is proved that for continuously differentiable functions $f_1(x,y),f_2(x,y),\dots,f_n(x,y)$ a region $U$ of the $x$, $y$ plane can be found such that the dimension of the space of vectors $(\varphi_1(t),\dots,\varphi_n(t))$ for which $\sum_{i=1}^n\varphi_i(f_i(x,y))=0$ in $U$, where $\varphi_i(t)\in L_2$, either equals infinity or else does not exceed the number $(n-1)n/2$. Superpositions of the form $\sum_{i=1}^n\psi_i(f_i(x,y))$ are also shown to be closed and nowhere dense in $L_2$.
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