How to prove that a predicate is prefix closed

Question

Suppose we have the predicate

$\qquad A.p.q ≡ (∀i \mid p≤i≤j<q : X.i≤X.j)$

which says that $X[p..q)$ is ascending.

Apparently, the predicate holds for empty segments, is prefix closed and is postfix closed.

I would like to prove that the predicate above is indeed prefix closed, but I am not able to; that is, I am unable to prove that

$\qquad A.p.q \implies (∀s \mid p≤s≤q : A.p.s)$

and I wondered if somebody could help?