Completeness theorem in $(q_1,q_2)$-quasimetric spaces

In $(q_1,q_2)$-quasimetric space $(X,d)$ we proved the completeness theorem for $(q_1,q_2)$-quasimetric space $(\mathcal{M}_{\overline{d}},H)$, where $\mathcal{M}_{\overline{d}}$ is the set of all $\overline{d}$-closed sets, $\overline{d}$ is conjugate to $d$ $(q_2,q_1)$-quasimetric, $H$ is the Hausdorff distance.