Question: Why is the face of a simple polytope a simple polytope?
My reasoning thus far:
I only have to show that this is true for any facet $F$ of $P$. Now let's take $v$ a vertices of $F$ and we show that $v$ is in $d-1$ facets of $F$.
Now take $F'$ a facet of $F$ containing $v$. Then by the diamond property, there exists a unique facet $G$ of $P$ containing $v$ such that $F'$ is in $G$ and $F$. Therefore the number of facets of $F$ containing $v$ is smaller or equal to $d-1$ (since there are only $d-1$ facets of $P$ containing $v$ other than $F$).
However I don't see how I can show that it cannot be strictly smaller than $d-1$