Let $G = (V,E)$ be a connected, undirected graph with a real-valued weight function $w$ defined on $E$. Let $A$ be a subset of $E$ that is included in some minimum spanning tree for $G$, and let $C = (V_C,E_C)$ be a connected component (tree) in the forest $G_A = (V,A)$. If $(u,v)$ is a light edge connecting $C$ to some other component in $G_A$, then $(u,v)$ is safe for $A$.
Proof The cut $(V_C,V-V_C)$ respects $A$, and $(u,v)$ is a light edge for this cut. Therefore, $(u,v)$ is safe for $A$. $\quad\blacksquare$
I did understand the original proof for the Cut property in CLRS (Theorem 23.1). But I am unable to grasp how does the above corollary follow from the original proof? Why does the cut $(V_C,V-V_C)$ respect $A$?