A property of dominance of partitions

Given an integer partition $\text{la}=({\text{la}}_1,\mathrm{.}\mathrm{.}\mathrm{.},{\text{la}}_{\mathrm{\ell }})$ and an integer k, denote by ${\text{la}}^{(k)}$ the sequence of length $\mathrm{\ell }$ obtained by reordering the values $\mathrm{\mid }{\text{la}}_i-k\mathrm{\mid }$ in non-increasing order. If $\text{la}$ dominates $\mu$ and has the same weight, then ${\text{la}}^{(k)}$ dominates ${\mu }^{(k)}$.