Help with proof about quotient groups

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Suppose $G$ and $H$ are groups, $N \triangleleft G$ and $K\triangleleft H$ are subgroups. Show that $N\times K\triangleleft G\times H$ and $(G\times H)/(N\times K)\cong(G/N)\times(H/K)$.

I was in the middle of constructing a isomorphism, but I stopped since to me it seems that $(G\times H)/(N\times K)$ is exactly $(G/N)\times(H/K)$, is it enough to show that they are equal by double containment or is an isomorphism better to do?

And if the isomorphism route is more viable, I'm stuck with

$f: (g, h) \times (N, K) \to (gN, hK)$, which is where my initial confusion stems from. I have an idea to use conjugation but I want to know whether I'm headed in the right direction, if any.

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Well, these quotients are not literally the same: the elements of $G\times H/(N \times K)$ are subsets of $G\times H$, while the elements of $(G/N)\times (H/K)$ are pairs of cosets.

I think the cleanest approach is to apply the first isomorphism theorem to $G\times H\to (G/N)\times (H/K)$.