the elements of the MIB $(p_1(x),\dots,p_q(x))$ of $G$ define a set of $K$-invariant polynomial functions of $v=(x_1,\dots ,x_\nu)\in V$. Therefore, by the Hilbert theorem recalled in the previous section, they can be expressed as polynomial functions of $\lambda$, that is,

[$${p\big|_V=\phi\circ\lambda.}\label{phi}$$]{} Let us remark that, in our assumptions, there are no relations among the elements of the MIB $\{\lambda\}$ and, consequently, possible relations $F_\alpha(p)=0$ among the $p_i$’s are identically satisfied for $p=\phi(\lambda)$.

From [(\[phi\])]{} and [(\[ins\])]{}, one immediately obtains

[$${\overline{\widehat S}\,=\,p(V)\,=\,\phi(V/K)\,\supset\,
\phi(\widehat\Sigma)\,\supseteq\,\phi(\widehat{\mathcal
V})\,=\,p({\mathcal V})=\widehat S}\label{pins}$$]{} and, since $\widehat\Sigma$, being a principal stratum, is the set of interior points of $V/K$ and is connected, $\phi(\widehat\Sigma)$ will coincide with the set of interior points of the closure $\overline{\widehat S}$ of $\widehat S$ and will be connected. This set does not coincide with $\widehat S$ if $\overline{\widehat S}$ contains, in its interior, points representing bordering strata of $\widehat S$. This certainly happens if the set $\widehat S$ is not connected and, presumebly, also if it is not multiply connected[^3].

The identification of points $\lambda\in\widehat \Sigma$, if any, whose image $\phi(\lambda)\not\in\widehat S$, can be obtained in the following way.

Let $x=v\oplus v_\bot$ yield the decomposition of $x\in\real^n$ in its vector components $v\in V$ and 