Proposition 8

Let $R$ be a ring, $(B_\lambda)_{\lambda \in L}$ a family of subrings of $R$ and for each $\lambda \in L$ let $A_\lambda$ be a subring of $B_\lambda$. If each $A_\lambda$ is integrally closed in $B_\lambda$, then $A = \bigcap_{\lambda \in L} A_\lambda$ is integrally closed in $B = \bigcap_{\lambda \in L} B_\lambda$.

This follows immediately from Definition 3 and no. 1, Corollary 1 to Proposition 2.