Cells in hyperspaces

Let C ( X ) denote the hyperspace of subcontinua of a continuum X. For A ∈ C ( X ) , define the hyperspace C ( A , X ) = { B ∈ C ( X ) : A ⊂ B } . Let k ∈ N , k ⩾ 2 . We prove that A is contained in the core of a k-od if and only if C ( A , X ) contains a k-cell.