[proofplan] The statement reduces immediately to the previous theorem once we observe that the standard $n$-simplex, viewed as the simplicial complex consisting of $\Delta^n$ together with all its faces, is a cone on any of its vertices. We verify this claim structurally and then invoke [Homology of a Cone](/theorems/1927) to conclude. [/proofplan]

[step:Identify $L$ as a cone on any vertex] Let the vertices of the standard $n$-simplex be $v_0, v_1, \dots, v_n$, so that $\Delta^n = [v_0, v_1, \dots, v_n]$. The simplicial complex $L$ consists of $\Delta^n$ together with all its faces — that is, every subset of $\{v_0, \dots, v_n\}$ spans a simplex of $L$. We claim that $L$ is a cone with cone point $v_0$ (the choice of $v_0$ is immaterial — the same argument works for any vertex). Recall that a simplicial complex $K$ is a cone on a vertex $v$ if, for every simplex $\sigma = [w_0, \dots, w_p]$ of $K$ with $v \notin \{w_0, \dots, w_p\}$, the coned simplex $[v, w_0, \dots, w_p]$ is also a simplex of $K$. Let $\sigma = [w_0, \dots, w_p]$ be a simplex of $L$ with $v_0 \notin \{w_0, \dots, w_p\}$. Then $\{w_0, \dots, w_p\} \subseteq \{v_1, \dots, v_n\}$, so $\{v_0, w_0, \dots, w_p\} \subseteq \{v_0, v_1, \dots, v_n\}$. Since every subset of $\{v_0, \dots, v_n\}$ spans a simplex of $L$, the set $\{v_0, w_0, \dots, w_p\}$ spans the simplex $[v_0, w_0, \dots, w_p]$ in $L$. This verifies the cone condition at $v_0$. [/step]

[step:Apply the homology of a cone] By the previous step, $L$ is a cone with cone point $v_0$. By [Homology of a Cone](/theorems/1927), \begin{align*} H_k(L) &= \begin{cases} \mathbb{Z} & k = 0, \\ 0 & k > 0, \end{cases} \end{align*} which is the stated result. This completes the proof. [/step]