Locally finite groups containing direct products of dihedral groups

Let $d$ be a fixed natural number. We prove the following: THEOREM. Let $G$ be a locally finite group saturated with groups from a set $\mathfrak{M}$ consisting of direct products of $d$ dihedral groups. Then $G$ is a direct product of $d$ groups of the form $B\leftthreetimes\langle v\rangle$, where $B$ is a locally cyclic group inverted by an involution $v$.