Equivalence of Definitions of Convex Polyhedron

$(1)$ implies $(2)$
Let $P$ be a convex polyhedron by definition $1$.

Thus $P$ is a convex polyhedron by definition $2$.

$(2)$ implies $(1)$
Let $P$ be a convex polyhedron by definition $2$.

Thus $P$ is a convex polyhedron by definition $1$.