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The Borsuk-Sieklucki theorem says that for every uncountable family ${X_{\alpha }}_{\alpha \in A}$ of n-dimensional closed subsets of an n-dimensional ANR-compactum, there exist α ≠ β such that $dim\left(X_{\alpha }\cap X_{\beta }\right)=n$. In this paper we show a cohomological version of that theorem: Theorem. Suppose a compactum X is $cl{c}_{\mathbb{Z}}^{n+1}$, where n ≥ 1, and G is an Abelian group. Let ${X_{\alpha }}_{\alpha \in J}$ be an uncountable family of closed subsets of X. If $di{m}_{G}X=di{m}_G{X}_{\alpha }=n$ for all α ∈ J, then $di{m}_G\left(X_{\alpha }\cap X_{\beta }\right)=n$ for some α ≠ β. For G being a countable principal ideal domain the above result was proved by Choi and Kozlowski...