# Why does the cut $(V_C,V-V_C)$ respect $A$?

Corollary 23.2

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 $$A$$?

The cut $$(V_C,V-V_C)$$ respects $$A$$ since $$C$$ is a connected component in $$(V,A)$$.