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In this paper, we prove that for every Finsler $(S^n, F)$ for $n\\geq3$ whose metric is induced by irreversible Finsler $(\\mathbb{R}P^n,F)$ with reversibility $\\lambda$ and flag curvature $K$ satisfying $(\\frac{\\lambda}{\\lambda+1})^2<K\\leq 1$, there exist at least $n-1$ prime closed geodesics on $(S^n, F)$. 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In this paper, we prove that for every Finsler $(S^n, F)$ for $n\\geq3$ whose metric is induced by irreversible Finsler $(\\mathbb{R}P^n,F)$ with reversibility $\\lambda$ and flag curvature $K$ satisfying $(\\frac{\\lambda}{\\lambda+1})^2<K\\leq 1$, there exist at least $n-1$ prime closed geodesics on $(S^n, F)$. 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