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Let $\Omega \subset {ℝ}^n$ be a compact semianalytic set and let $ℱ$ be a collection of real analytic functions defined in some neighbourhood of $\Omega$. Let $Y_{\omega }$ be the germ at $\omega$ of the set ${\bigcap }_{f\in ℱ}{f}^{-1}\left(0\right)$. Then there exist analytic functions $v_1,v_2,...,v_s$ defined in a neighbourhood of $\Omega$ such that $\frac{1}{2}\chi \left(\mathrm{lk}(\omega ,Y_{\omega })\right)={\sum }_{i=1}^s\mathrm{sgn}{v}_i\left(\omega \right)$, for all $\omega \in \Omega$.