\operatorname{ost}_{\beta L} \mathcal{L}$ defined by $C(F) = \operatorname{ost}_{\beta L} L_{i_F}$ is a carrier. Since $\mathcal{K}$ and $\mathcal{L}$ are isomorphic, the map $C' \colon \mathcal{F} \to \operatorname{ost}_{\beta K} \mathcal{K}$ defined by $C'(F) = \operatorname{ost}_{\beta K} K_{i_F}$ is a carrier as well. By $(\ast)$, $\varphi$ is carried by $C'$. By Corollary \[cor:open star swelling\], $\operatorname{ost}_{\beta K} \mathcal{K}$ is an $AE(n)$-cover. By the Carrier Theorem, $\varphi$ extends to a map $\tilde \varphi \colon B^{m+1} \to K$ that is carried by $C'$. This is a null-homotopy of $\varphi$ in $K$ and we are done with the proof that $p$ induces monomorphisms on homotopy groups of dimensions less than $n$.

Next we will show that $p$ induces epimorphisms on homotopy groups of dimensions less than $n$. Let $m < n$. Let $\psi \colon S^m \to L$. Let ${{\mathcal{G}_{}}}$ be a closed finite cover of $S^m$ that refines $\psi^{-1}(\operatorname{ost}_{\beta L} \mathcal{L})$. For each $G \in {{\mathcal{G}_{}}}$ pick $i_G \in I$ such that $\psi(G) \subset \operatorname{ost}_{\beta L} L_{i_G}$. Then the map $D \colon {{\mathcal{G}_{}}} \to \operatorname{ost}_{\beta L} \mathcal{L}$ defined by $D(G) = \operatorname{ost}_{\beta L} L_{i_G}$ is a carrier and $\psi$ is carried by $D$. Since $\mathcal{K}$ and $\mathcal{L}$ are isomorphic, the map $D'(G) = \operatorname{ost}_{\beta K} K_{i_G}$ 