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Let $\\cH=(V,\\cE_2\\cup \\cE_3\\cup\\dots\\cup\\cE_k)$ be a non-uniform hypergraph with the vertex set $V$ and the set $\\cE_i$ of edges of size $i=2,\\ldots , k$. Suppose that $\\cH$ has no $2$-cycles (regardless of sizes of edges), and neither contains $3$-cycles nor $4$-cycles consisting of $2$-element edges. 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