Show $(G\times H)/(K\times K') \cong G/K\times H/K'$ if $K\lhd G$ and $K'\lhd H$

Question

As the title says, I want to show that $(G\times H)/(K\times K') \cong G/K\times H/K'$ if $K\lhd G$ and $K'\lhd H$.

I already showed that $K\times K'$ is a normal subgroup of $G\times H$ and I think that I should use the isomorphism theorem stating that if $A,B$ are groups and $f:A\to B$ is a homomorphism then $f(A)=A/\text{ker}(f)$. But I can't figure out how to go further. Can you help me to prove this problem?

I finished proving using the idea from the comment.

Answer 1

An element of $(G \times H)/(K \times K')$ looks like $(g,h) \cdot (K \times K')$ for some $g \in G, h \in H$.
An element of $G/K \times H/K'$ looks like $(gK, hK')$, for some $g \in G, h \in H$.

Write down the map $f: (g,h) \cdot (K \times K') \mapsto (gK, hK')$. Show this map is well-defined, and that its an isomorphism. Then use the 1st Isomorphism Theorem and you're off to the races.