satisfiability – How to prove that $overline{L}$ is a strong connected component of an implication graph

Task:
Let $G$ be an implication graph.
Suppose that a set $L$ of literals is a strong connected component of $G$. Show that the set $overline{L} = {overline{l}space vert space l in L}$ is also a strong connected component of $G$. (Hint: Show by induction that for every path from $l$ to $l′$ a path from $overline{l′}$ to $l$ exists).

My Question:
I know that $overline{L}$ must also be a strong connected component of $G$, but how can I prove it using induction? Or is there another way to prove it?