An inequality of graph coloring

Question

Let $G_1=(V,E_1)$ and $G_2=(V,E_2)$ be two graphs with the same vertices. Define a new graph $G=(V,E_1 \cup E_2)$. Prove that $$χ(G)\leqslant χ(G_1)χ(G_2),$$ when $χ(G)$ is the coloring number of $G$.

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