} \prod_{q=0}^{p} \left[a_q^{(i)}, b_q^{(i)}\right]  \subset \bigcup_{k\in K_{p,{\mathfrak{m}}}} \prod_{q=0}^p \left[  \theta_q^{(i,\ell_{k})},\theta_q^{(i,\ell_{k})} + \varepsilon_{q}^{(i,\ell_k)}  \right].\end{aligned}$$ Let $