Proof. $(2) \Rightarrow (1)$: Let $\fB \subset \cn_{E}(0)$ be a fundamental system of neighbourhoods at $0$ consisting of convex, circled, and radial sets, then $\ol{\fB}= \bracsn{\ol U|U \in \fB}$ is a fundamental system of neighbourhoods at $0$ consisting of barrels.

$(2) \Rightarrow (3)$: Let $\rho: E \to [0, \infty)$ be a lower semicontinuous seminorm, then $\bracs{\rho > 1}$ is open and $\bracs{\rho \le 1}$ is a Barrel. In which case, $\rho$ is continuous by (4) of Lemma 11.1.9.

$(3) \Rightarrow (2)$: Let $D \subset E$ be a barrel and $\rho: E \to [0, \infty)$ be its gauge. By (4) of Definition 11.1.11, $D = \bracs{\rho \le 1}$, so $\bracs{\rho > 1}$ is open, and $\rho$ is semicontinuous. By assumption, $\rho$ is continuous, so $D \in \cn_{E}(0)$ by (5) of Lemma 11.1.9.$\square$