The proof that P ::== any sub-graph, G* of the tree G, is also a tree, involves proof by contradiction.
We can suppose that the sub-graph has a cycle --> the whole graph has a cycle --> the whole graph is a tree --> trees can't have a cycle --> contradiction --> therefore, the sub-graph does not have a cycle --> therefore, the sub-graph must also be a tree
however, another property of a tree is that it is connected. is our first proof sufficient for proving P or do we need to write another proof that talks about cycles?
I guess my real question is about how finding a contradiction can actually prove a proposition when you are only showing the contradiction in one of many possible properties (i.e. acyclic, rather than connected)