  For analytic map germs $f: (\mathbb{R}^n, 0)\to (\mathbb{R}, 0)$ having an
isolated critical value in the origin with $\dim V(f)>0$ and satisfying the
transversality property of D.B. Massey we show that for $c>0$ a large enough
constant, and $k$ a large enough natural number the local Milnor-L\^e fibers of
$f$ and of the isolated singularity $g=f-c(\sum_{i=1}^n x_i^2)^k$ satisfies the
following. There exists a homotopy equivalence between the negative Milnor-L\^e
fiber of $f$, to which a cobordism between its boundary and the link of $f$ has
been adjoined, and the negative Milnor-L\^e fiber of $g$.
