\paragraph{Problem 4 (18 points)}

Let $F:=\{ F^n:=\{ f^n_k: \{0,1\}^n \to \{0,1\}^n  \}_{k\in \{0,1\}^{n}} \}_{n\in\mathbb{N}}$ be a family of PRFs. 
Let $G:=\{ G^n:=\{ g^n_j: \{0,1\}^n \to \{0,1\}^n  \}_{j\in \{0,1\}^{n}} \}_{n\in\mathbb{N}}$ be another family of PRFs. 

Let $H:=\{ H^n:=\{ h^n_{s,j}: \{0,1\}^n \to \{0,1\}^n  \}_{s\in\{0,1\}^n, j\in\{0,1\}^n } \}_{n\in\mathbb{N}}$ be defined by $h^n_{s,j}(x) = f^n_{g^n_j(x)}(s)$. Prove that $H$ is a PRF family.