  Let $G$ be a real linear algebraic group and $K$ be a maximal compact
subgroup. Let $\mathfrak{p}$ be a standard minimal parabolic subalgebra of the
complexified Lie algebra $\mathfrak{g}$ of $G$. In this paper we show that: for
any moderate growth smooth Fr\'echet representation $V$ of $G$, the map
$V_{K}\subset V$ induces isomorphisms $H_{i}(\mathfrak{n},V_{K})\cong
H_{i}(\mathfrak{n},V)$ ($\forall i\geq 0$). This is called Casselman's
comparison theorem in the literature. As a consequence, we show that: for any
$k\geq 1$, $\mathfrak{n}^{k}V$ is a closed subspace of $V$ and the inclusion
$V_{K}\rightarrow V$ induces an isomorphism
$V_{K}/\mathfrak{n}^{k}V_{K}=V/\mathfrak{n}^{k}V$. This strengthens Casselman's
automatic continuity theorem.
