Equational Noetherianness of rigid soluble groups

A group $G$ is said to be rigid if it contains a normal series of the form
$$ G=G_1>G_2>\dots>G_m>G_{m+1}=1, $$
whose quotients $G_i/G_{i+1}$ are Abelian and are torsion free as right $Z[G/G_i]$-modules. In studying properties of such groups, it was shown, in particular, that the above series is defined by the group uniquely. It is known that finitely generated rigid groups are equationally Noetherian: i.e., for any $n$, every system of equations in $x_1,\dots,x_n$ over a given group is equivalent to some of its finite subsystems. This fact is equivalent to the Zariski topology being Noetherian on $G^n$, which allowed the dimension theory in algebraic geometry over finitely generated rigid groups to have been constructed. It is proved that every rigid group is equationally Noetherian.