On independent families of normal subgroups in free groups

Consider a presentation $\mathcal{P}=\Bigl\langle\mathbf x\mid \bigcup\limits_{i=1}^n \mathbf r_i\Bigr\rangle$. Let $\mathbf R_i$ be the normal closure of the set $\mathbf r_i$ in the free group $\mathbf F$ with basis $\mathbf x$, $\mathcal{P}_i=\langle \mathbf{x}\mid\mathbf r_i\rangle$, $\mathbf N_i = \prod\limits_{j\neq i}\mathbf R_j$. In this paper, using geometric techniques of pictures, generators for $\frac{\mathbf R_i\cap \mathbf N_i}{[\mathbf R_i, \mathbf N_i]}$, $i=1,\ldots,n$, are obtained from a set of generators over $\{\mathcal P_i\mid i=1,\ldots, n\}$ for $\pi_2(\mathcal{P})$. As a corollary, we get a sufficient condition for the family $\{\mathbf R_1,\ldots,\mathbf R_n\}$ to be independent.