  Given pure-dimensional (generalized) cycles $\mu_1$ and $\mu_2$ on a complex
manifold $Y$ we introduce a product $\mu_1\diamond_{Y} \mu_2$ that is a
generalized cycle whose multiplicities at each point are the local intersection
numbers at the point. % If $Y$ is projective, then given a very ample line
bundle $L\to Y$ we define a product $\mu_1\bl \mu_2$ whose multiplicities at
each point also coincide with the local intersection numbers. In addition,
provided that $\mu_1$ and $\mu_2$ are effective, this product satisfies a
B\'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 of
St\"uckrad-Vogel cycles on $\Pk^N$. There are quite explicit relations between
$\di_Y$ and $\bl$.
