Lemma 8

a) Let $U = (u_{ij}) \in \mathbf{SL}(n, K)$. In order that $U \in \Omega'$, it is necessary and sufficient that $\det(u_{ij})_{k \leq i, j \leq n} \neq 0$ for $k = 2, 3, \ldots, n$.

b) $\Omega'$ is an open subset of $\mathbf{SL}(n, K)$.

c) The mapping $\psi$ is a homeomorphism of $T_1(n, K) \times G'_1$ onto $\Omega'$.

For, let $M \in T_1(n, K)$ and $N \in {}^t(T(n, K)^*)$. In order that $M \cdot N \in \mathbf{SL}(n, K)$, it is necessary and sufficient that $N \in G'_1$. Therefore $\Omega' = \mathbf{SL}(n, K) \cap \Omega$ and Lemma 8 follows at once from Lemma 7.