Let $d\ge1$. A $d$-simplex $S$ is the convex hull in $\mathbb R^d$ of the vertices $v_0,\dots,v_d\in\mathbb R^d$ where $\{v_1-v_0,\dots,v_d-v_0\}$ is a linearly independent set of $d$ vectors; for any non-empty subset $T$ of vertices, their convex hull is a face of $S$ and it is a facet of $S$ if $|T|=d$.

Let $S$ be a $d$-simplex. Let $m\ge1$. Let $S_1,\dots,S_m$ be $d$-simplices such that for $1\le i<j\le m$, $S_i\cap S_j$ is either empty or a face of $S_i$ and $S_j$. Assume that $S=\bigcup_{i=1}^m S_i$.

Let $i\in\{1,\dots,m\}$. Show that if a facet of $S_i$ is a subset of a facet of $S$, then it is not a face of any other $S_j$ for $j\in\{1,\dots,m\}$; otherwise, it is a face of exactly one other $S_j$ for some $j\in\{1,\dots,m\}$.

You may use, if needed, the solution to A lemma on intersections of $d$-simplices