Generating the kernel of a staircase starshaped set from certain staircase convex subsets

Let S be an orthogonal polygon in the plane. Assume that S is starshaped via staircase paths, and let K be any component of Ker S, the staircase kernel of S, where K ≠ S. For every x in S\K, define W _K (x) = {s: s lies on some staircase path in S from x to a point of K}. There is a minimal (finite) collection W(K) of W _K (x) sets whose union is S. Further, each set W _K (x) may be associated with a finite family U _K (x) of staircase convex subsets, each containing x and K, with ∪{U: U in U _K (x)} = W _K (x). If W(K) = {W _K (x ₁), ..., W _K (x _n )}, then K ⊆ V _K ≡ ∩{U: U in some family U _K (x _i ), 1 ≤ i ≤ n} ⊆ Ker S. It follows that each set V _K is staircase convex and ∪{V _k : K a component of Ker S} = Ker S.