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In this paper, the performance of a wireless system, which is beneficial from utilization of multiple reconfigurable intelligent surfaces (RISs) and the direct link between the source node and the mobile user, is analyzed. Especially, the effect of transceiver hardware impairments (HIs) are taken into consideration. We successfully derive the outage probability (OP), throughput, achievable data rate (ADR), and symbol error probability (SEP) expressions of this multiple-RIS assisted wireless systems with and without HIs (referred as RIS-HI and RIS-ID system, respectively) over Nakagami-<inline-formula> <tex-math notation="LaTeX">m </tex-math></inline-formula> fading channels. The above performance factors are also benchmarked with the case that only a point-to-point (P2P) link without RIS is considered (denotes as as P2P-HI and P2P-ID systems for HI and ideal hardware conditions, respectively). All obtained formulas are verified by Monte-Carlo simulations. It is observed that the RIS-HI system significantly outperforms the P2P-HI system at low transmit power regime, while the performance of the RIS-ID system is always better than that of P2P-ID system. On the other hand, HIs show more severe effect for higher data transmission rates or high modulation orders.