Therefore, there is only one simple path from v to w in T . Every two vertices are connected by a simple path. Add a new edge, say from x to y . This creates a circuit because x and y are already connected in T . If more than one circuit was created, then x and y would have to be joined by two or more paths in T , which is not the case, so only one circuit is created.
T has no circuits, but the addition of any edge creates exactly one circuit.
Must prove that T is connected. Let v and w be any vertices of T . If no edge exists from v to w then add such an edge and you obtain a circuit. If you then remove this edge, then you'll be left with a path from x to y in T , so T is connected. This completes the proof of the theorem.