To show this, let's take the Euclidean space of step (b) and consider a path between the origin and any of its representatives. This path consists in a series of edges (u, v, t) ∈ V ×V × 3 where t is the translation between vertex v and the vertex v ′ which is the actual neighbour of u. After any number of steps, let's note (u, τ) ∈ V × 3 the pair where u is identifier of the vertex on which the path has just landed and τ is the sum of the translations t encountered so far. The initial pair thus is (1, 0 3 ), since 1 is the number of the origin. An induction on the number of steps shows that the position of the exact vertex on which the path lands when reaching pair (u, τ) is that of vertex u translated by τ. As a consequence, when reaching the final vertex of the path, the pair is (u end , τ end ) = (1, τ end ). Thus, by construction τ end is the sum of translations t stored in the edges during the algorithm, so it is an integer combination of the positive translations of the set. But since this reasoning was valid for the translation between the origin and any of its representatives, this proves that any valid translation can be written as an integer combination of positive translations of the set.