I am used to the following definition of a (proper) face of a polytope:
A nonempty convex subset $F$ of a polytope $C$ is called a face of $C$ if $\alpha x + (1-\alpha) y \in F$ with $x, y \in C$ and $0 < \alpha < 1$ imply $x, y \in F$. $F$ is a proper face of $C$ if $F$ is a face of $C$ and $F \neq C$.
I often read the following definition of a proper face:
A nonempty subset $F$ of a polytope C is a proper face of $C$ if there is closed half-space $H$ containing $C$ such that $F = C \cap \partial H$, where $\partial H$ is the affine hyperplane defined by the boundary of $H$.
How can I show that these two definitions are equivalent?