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For $j\\in \\{1,2,\\ldots,m\\}\\backslash\\{i\\}$, let $\\mathscr{G}_j:=\\{A\\in\\mathscr{G}: j\\in A\\}$ and $\\mathscr{F}_j:=\\{A\\in\\mathscr{F}: j\\in A\\}$. In this note, we will prove a lemma which says that if $\\frac{|\\mathscr{G}_j|}{|\\mathscr{G}|}\\geq c\\,(c\\in (0,1])$, then $\\frac{|\\mathscr{F}_j|}{|\\mathscr{F}|}\\geq \\frac{1}{1+2(1-c)/c}$. 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