Consider a set of $m$ points in $\mathbb{R}^n$, $2 \le m \le {n \choose 2}$. We do not know the coordinates of the points, but we know the distances of each point from any other point. However, for each point $P_i, i=1, \ldots ,m$, we just have a multiset $Q_i$ of $m - 1$ distances, without knowing which other point every distance value refers to.
Two sets, of $m$ points each, are isomorphic if we can apply a rigid transformation (rotation and/or translation and/or reflection) so that each point of the second set will be in the same position of one point of the first set.
Defined the multiset $S_i, i=1, \ldots ,m$, for the second set as we have defined $Q_i$ for the first set, is this condition:
$\{S_1, \ldots , S_m\} = \{Q_1, \ldots, Q_m\}$
a sufficient condition to say that the two sets of points are isomorphic?