The variety $\mathfrak{A}_{m,n}$ is defined by the system of $n$-ary operations $\omega_1,\dots,\omega_m$, the system of $m$-ary operations $\varphi_1,\dots,\varphi_n$, $1\leqslant m\leqslant n$, and the system of identities $$ \begin{aligned} x_1\dots x_n\omega_1\dots x_1\dots x_n\omega_m\varphi_i &=x_i \qquad (i=1,\dots,n),\\ x_1\dots x_m\varphi_1\dots x_1\dots x_m\varphi_n\omega_j &=x_j \qquad (j=1,\dots,m).\\ \end{aligned} $$ It is proved in this paper that the subalgebra $U$ of the free product $\prod_{i\in I}^*A_i$ of the algebras $A_i$ ($i\in I$) can be expanded as the free product of nonempty intersections $U\cap A_i$ ($i\in I$) and a free algebra.