Let $G_1$ and $G_2$ be two groups.
Let $H_1$ and $H_2$ be normal subgroups of $G_1$ and $G_2$ respectively.
Then prove that $(G1\times G2)/(H1\times H2)$ is isomorphic to $(G1/H1)\times(G2/H2)$.
You can consider the obvious surjective homomorphism $G_1\times G_2\to (G_1/H_1)\times (G_2/H_2)$. What is its kernel? Then you can use the first isomorphism theorem. This is essentially the same solution as if you just defined a map and checked that it is an bijective homomorphism. But using the 1st isom. theorem you don't need to do this many verifications, they are already encoded.