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Given pure-dimensional (generalized) cycles $\mu_1$ and $\mu_2$ on a complexmanifold $Y$ we introduce a product $\mu_1\diamond_{Y} \mu_2$ that is ageneralized cycle whose multiplicities at each point are the local intersectionnumbers at the point. % If $Y$ is projective, then given a very ample linebundle $L\to Y$ we define a product $\mu_1\bl \mu_2$ whose multiplicities ateach point also coincide with the local intersection numbers. In addition,provided that $\mu_1$ and $\mu_2$ are effective, this product satisfies aB\'ezout inequality. If $i\colon Y\to \Pk^N$ is an embedding such that$i^*\Ok(1)=L$, then $\mu_1\bl \mu_2$ can be expressed as a mean value ofSt\"uckrad-Vogel cycles on $\Pk^N$. There are quite explicit relations between$\di_Y$ and $\bl$.

On non-proper intersections and local intersection numbers