Exercise 3, § 3

Let G, H be two graphs. Show that the the relation $\mathrm{pr}_1\mathrm{H} \subset \mathrm{pr}_1\mathrm{G}$ is equivalent to $\mathrm{H} \subset \mathrm{H} \circ \overset{-1}{\mathrm{G}} \circ \mathrm{G}$. Deduce that $\mathrm{G} \subset \mathrm{G} \circ \overset{-1}{\mathrm{G}} \circ \mathrm{G}$.