shape loop space of pro-discrete spaces

In this paper, considering the kth shape loop space ˇω p k (x, x), for an hpol∗-expansion p : (x, x) → ((xλ, xλ), [pλλ′ ], λ) of a pointed topological space (x, x), first we prove that ˇωk commutes with the product under some conditions and then we show that ˇωp k (x, x) ∼= lim ← ˇω p k (xi, xi), for a pro-discrete space (x, x) = lim ← (xi, xi) of compact polyhedra. finally, we conclude that these spaces are metric, second countable and separable.