MathNet 1981 — Part 01 Question 8

$P(x)$ and $Q(x)$ are two polynomials that satisfy the identity $P(Q(x)) \equiv Q(P(x))$ for real numbers $x$. If the equation $P(x)=Q(x)$ has no real solution, show that the equation $P(P(x))=Q(Q(x))$ also has no real solution.