Rationality of the vertex operator algebra $V_L^{+}$ for a positive definite even lattice $L$

The lattice vertex operator algebra $V_L$ associated to a positive definite even lattice $L$ has an automorphism of order 2 lifted from -1-isometry of $L$. We prove that for the fixed point vertex operator algebra $V_L^{+}$, any ${\mathbb{Z}}_{\ge 0}$-graded weak modules is completely reducible.