d}}(x_1,\dots ,x_{lk})).$$ These terms come from $\sigma\in\prod_{=1,k,\dots ,k}\subset\prod_{k,\dots ,k} ,$ where $\prod_{=1,k,\dots ,k}$ is a singleton with the only partition containing $mk$ blocks all of cardinality one.$\Box $

Let $B\subset {\mathbb R}^d,\,d\in {\mathbb N}$ be as in Section 2, $X$ be a point process of compact sets [@RefS] called grains and we denote ${\mathbf x}$ realization of $X$ on $B$, i.e. the union of all compact sets. Consider that there is a probability density [@RefM] $$\label{den}p({\mathbf x})=c_\nu\exp(\nu\cdot G({\mathbf x})),$$of $X$ w.r.t. a given reference Poisson point process $\eta $ of compact sets. Here $\nu =(\nu_1,\dots ,\nu_d)$ is a vector of real parameters, $c_\nu $ is a normalizing constant, $$G({\mathbf x})=(G_1({{\mathbf x}}),\dots,G_d({{\mathbf x}}))\in {\mathbb R}^d$$ is a vector of geometrical characteristics of ${\mathbf x}.$ In the exponent of (\[den\]) there is the inner product in ${\mathbb R}^d.$ The largest set of ${\nu}$ such that exponential family density (\[den\]) is well defined is $$\{\nu\in\mathbb{R}^d : \mathbb{E}[\exp(\nu\cdot G(\eta ))] <\infty\}.$$ For $m\in\mathbb{N}$ denote $$D^m_{y_1,\dots,y_m}G({\mathbf x})=(D^m_{y_1,\dots,y_m}G_1({{\mathbf x}}),\dots,D^m_{y_1,\dots,y_m}G_d({{\mathbf x}}))^T$$ the vector of $m-$th differences.

\[Th:papan\]Consider the probability density (\[den\]). Then for the corresponding conditional intensity $\lambda^*_m$ of order $m\in {\mathbb N}$ and a realization ${\mathbf x}$ of $X$ it holds $$\lambda^*_m(y_1,\dots,y_m;{\mathbf x})=\exp (\nu\cdot Q_mG({\mathbf x})),$$ 