from $\mathcal{S}'$ by adding a $\otimes$ or $\parr$ hyperedge $e$, and if $d$ is above the left source (resp. the right source) of $e$.

-   $\ploc_\mathcal{S}(d,x)=\ploc_{\mathcal{S}'}(d,x)$ otherwise.

The partial address of $d$ is defined with respect to either a conclusion of the structure of the source of a cut hyperedge, which is uniquely defined as the occurrence of formula $c$ such that $d$ is above $c$ and $c$ is not source of either a $\otimes$ or a $\parr$; the *address of $d$* is then defined as the term $\loc_\mathcal{S}(d,x)=p_c(\ploc_\mathcal{S}(d,x))$.

Let $\mathcal{S}$ be a proof-structure. We write $\mathrm{Ax}(\mathcal{S})$ (resp. $\mathrm{Cut}(\mathcal{S})$) the set of axioms (resp. cut) hyperedges in $\mathcal{S}$. Given $e\in\mathrm{Ax}(\mathcal{S})$ ($e\in\mathrm{Cut}(\mathcal{S})$), we write $C_e^l$ and $C_e^r$ the left and right conclusions (resp. sources) of $e$ respectively.

\[def:vehicle\] The *vehicle* of proof-structure $\mathcal{S}$ is the constellation $\mathcal{S}^{\mathrm{ax}}$ defined by: $|\mathcal{S}^{\mathrm{ax}}| = \mathrm{Ax}(\mathcal{S})$ and $\alpha_{\mathcal{S}^{\mathrm{ax}}}(e) = \gstar{\loc_{\mathcal{S}}(C_e^l), \loc_{\mathcal{S}}(C_e^r)}$.

\[def:colouration\] The colouration $\pm c.\Sigma$ of $\Sigma$ with a colour $c \in \mathcal{C}$ and a polarity $\pm \in \{+, -\}$ changes all the rays of its stars by the following functions $\phi_c^\pm$ defined by: $\phi_c^\pm(+d.t) = \pm c.t \quad \phi_c^\pm(-d.t) = \pm c.t \quad \phi_c^\pm(t) = \pm c.t$.

Let $\Sigma_1,\Sigma_2$ be two constellations. We 