Introduction to graph theory/Proof of Corollary 3

Statement
A graph is a tree if and only if for every pair of distinct vertices $$u,v$$, there is exactly one $$u,v$$-path.

Corollary of
Theorem 2: A graph is a forest if and only if for every pair of distinct vertices $$u,v$$, there is at most one $$u,v$$-path.

Proof
A tree is defined to be a connected forest. Furthermore, a graph is defined to be connected if and only if for every pair of distinct vertices $$u,v$$, there is at least one $$u,v$$-path. The proof should now be evident, but for completeness:

If graph $$G$$ is not a tree, it is either not a forest, or not connected. If $$G$$ is not a forest, by Theorem 2, there exists a pair of vertices $$u,v$$ with more than one $$u,v$$-path. If $$G$$ is not connected, there exists a pair of vertices $$u,v$$ with no $$u,v$$-path. Thus if $$G$$ is not a tree, it is not the case that each pair of vertices has precisely one $$u,v$$-path.

If graph $$G$$ is a tree, fix a pair of vertices $$u,v$$. As G is a forest, by Theorem 2, there exists at most one $$u,v$$-path. Since $$G$$ is connected, there is at least one $$u,v$$-path. Thus there is exactly one $$u,v$$-path. Since we have proved this for any fixed pair of vertices $$u,v$$, it follows that if G is a tree, each pair of vertices has precisely one path.