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We prove that if all $V$-modules are completely reducible and a fixed point subVOA $V^\\sigma$ is $C_2$-cofinite, then all $V^\\sigma$-modules are completely reducible and every simple $V^{\\sigma}$-module appears in some twisted or ordinary $V$-modules as a $V^{\\sigma}$-submodule. We also prove that $V_L^{\\sigma}$ is $C_2$-cofinite for any lattice VOA $V_L$ and $\\sigma\\in \\Aut(V_L)$ lifted from any triality automorphism of $L$. 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