The $r$-fold analogues of Whitney trick were `in the air' since 1960s. However, only in this century they were stated, proved and applied to obtain interesting results. Here we prove and apply a version of the $r$-fold Whitney trick when general position $r$-tuple intersections have positive dimension. Theorem. Assume that $D=D_1\sqcup\ldots\sqcup D_r$ is disjoint union of $k$-dimensional disks, $rd\ge (r+1)k+3$, and $f:D\to B^d$ a proper PL (smooth) map such that $f\partial D_1\cap\ldots\cap f\partial D_r=\emptyset$. If the map $$f^r:\partial(D_1\times\ldots\times D_r)\to (B^d)^r-\{(x,x,\ldots,x)\in(B^d)^r\ |\ x\in B^d\}$$ extends to $D_1\times\ldots\times D_r$, then there is a proper PL (smooth) map $\overline f:D\to B^d$ such that $\overline f=f$ on $\partial D$ and $\overline fD_1\cap\ldots\cap \overline fD_r=\emptyset$.
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