note the composition $\hat h \circ \lambda: N^{n-2k}_{ext} \to \hat X \to
\R^n$ by $g_1$. Let us denote the mapping $ \hat h \circ \psi_Q: Q \to \hat X \to \R^n$ by $
\varphi_Q$. One can assume that the mapping $\varphi_Q$ is an embedding. Moreover, without loss of generality one may assume that this embedding is extended to a generic embedding $\varphi_P: P \subset
\R^n$ such that the embedded polyhedron $\varphi_P: P \subset
\R^n$ does not intersect $g_1(N^{n-2k}_{ext})$.

Let us denote by $U_{\varphi}(P)$ a regular neighborhood of the subpolyhedron $\varphi_P(P) \subset \R^n$ (we may assume that the radius of this neighborhood is equal to $\varepsilon$). Up to an isotopy a regular neighborhood $U_{\varphi}(P)$ is well-defined, in particular, this neighborhood does not depend on the choice of a regular embedding of $P$, moreover $U_{\varphi}(P)$ and $U(P)$ are diffeomorphic.

Without loss of generality after an additional small deformation we may assume that the restriction $g_1
\vert_{N^{n-2k}_{int}}$ is a regular immersion $g_1:
N^{n-2k}_{int} \subset \R^n$ with the image inside $U_{\varphi}(P)$. In particular, the restriction of $g_1$ to the boundary $N^{n-2k-1}_Q = \partial(N^{n-2k}_{int})$ is a regular embedding $N^{n-2k-1}_Q \subset \partial U(P)$. The immersion $g_1
\vert_{N_{int}}$ is conjugated to the immersion $f
\vert_{N_{int}}$ by means of a diffeomorphism of $U_{\varphi}(P)$ with $U(P)$.

By Proposition 3, for an arbitrary $\varepsilon_2 >0$, $\varepsilon_2 << \varepsilon_1 << \varepsilon$, there exists an immersion $g : N^{n-2k} \looparrowright \R^n$ in the regular homotopy class of $f$, such that $g$ coincides with $g'$ (and with $g_1$) on $N^{n-2k}_{int}$ and, moreover, $dist(g,g_1)<
\varepsilon_2$.

Let us consider the self-intersection manifold $L^{n-4k}$ of the immersion $g$. This manifold is a submanifold in $\R^n$. Let us construct the mappings $\kappa_a: L^{n-4k}
\to K(\Z/2,1)$, $\mu_a: L^{n-4k} \to K(\Z/4,1)$. Then we check the conditions (8) and (9).

The manifold $L^{n-4k}$ is naturally