The photo-neutron cross sections of $^{162,163}rm{Dy}$ have been measured for the first time in an energy region from the neutron threshold ($S_n$) up to $approx$ $13$~MeV. The ($gamma$,n) reaction was induced with quasi-monochromatic laser Compton-scattered $gamma$ rays, produced at the NewSUBARU laboratory. The corresponding $gamma$-ray strength functions ($gamma$SF) have been calculated from the photo-neutron cross sections. The data are compared to reanalyzed $gamma$SFs of $^{160-164}rm{Dy}$, which are measured below $S_n$. The excellent agreement with the photo-neutron data at $S_n$ confirms the principle of detailed balance. Thus, a complete $gamma$SF is established covering in total the energy region of 1 MeV $leq$ E$_{gamma}$ $leq$ 13 MeV. These mid-shell well-deformed dysprosium isotopes all show scissors resonances with very similar structures. We find that our data predict the same integrated scissors strength as ($gamma,gamma^prime$) data when integrated over the same energy range, which shows that the scissors mode very likely is consistent with the generalized Brink hypothesis. Finally, using the $gamma$SFs as input in the reaction code TALYS, we have deduced radiative neutron-capture cross sections and compared them to direct measurements. We find a very good agreement within the uncertainties, which gives further support to the experimentally determined $gamma$SFs.
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