Assume that $K$ is normal in $G$. We want to show that $K/H$ is normal in $G/H$. Let $gH \in G/H$ and $kH \in K/H$. We need to show that $(gH)(kH)(gH)^{-1} \in K/H$. Since $K$ is normal in $G$, we have $gkg^{-1} \in K$ for all $g \in G$ and $k \in K$. Thus,
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