It is proved that if $ n\geq2$, then $\chi_{at}(K_n\times K'_n)=2n$, and if $ p,\,q\geq2$, then $\chi_{at}(C_{2p}\times K_{2q})=2q+3$, where $K_n\times K'_n$ is the product of two complete graphs with different labels, and $C_{2p}\times K_{2q}$ is the product of an even cycle and a complete graph of even order.
