17.105 (2010)

An equation over a pro-$p$-group $G$ is an expression $v(x) = 1$, where $v(x)$ is an element of the free pro-$p$-product of $G$ and a free pro-$p$-group with basis $\{x_1, \dots, x_n\}$; solutions are sought in the affine space $G^n$. Is it true that a free pro-$p$-group is equationally Noetherian, that is, for any $n$ every system of equations in $x_1, \dots, x_n$ over this group is equivalent to some finite subsystem of it?