On the cardinality of relator sets of groups $F/\prod [N_i,N_j]$

The present paper generalizes the results of our paper "On the finite presentability of the group $F/[M,N]$" to an arbitrary family of normal subgroups $\{N_i \mid i\in I\}$ in a free group $F$. We obtain conditions for finite presentability of the quotient group $F/\prod [N_i,N_j]$. In both papers, the proof of the main result is based on the properties of verbal wreath products introduced by Alfred Lvovich Shmel'kin.