Corollary 1

Suppose that $g$ is a free $K$-module. Let $W$ be a sub-K-module of $T^{n}$.

If, in the notation of diagram (3), the restriction of $\tau_n$ to $W$ is an isomorphism of $W$ onto $S^n$, then the restriction of $\psi_n$ to $W$ is an isomorphism of $W$ onto a supplement of $U_{n-1}$ in $U_n$.

The restriction to $W$ of $\omega_n \circ \tau_n$ is a bijection of $W$ onto $G^n$; so is the restriction $\theta_n \circ \psi_n$ to $W$. Hence the corollary.