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It was recently found that cascading failures can cause the abrupt breakdown of a system of interdependent networks. Using the percolation method developed for single clustered networks by Newman (Phys. Rev. Lett., 103 (2009) 058701), we develop an analytical method for studying how clustering within the networks of a system of interdependent networks affects the system's robustness. We find that clustering significantly increases the vulnerability of the system, which is represented by the increased value of the percolation threshold p c in interdependent networks.