T is a tree. T has n - 1 edges and no circuits. T is connected and has n - 1 edges. T is connected and each of its edges is an isthmus. Any two vertices of T are connected by exactly one simple path.
T has no circuits, but the addition of any new edge creates exactly one circuit.
Note The theorem states that all of the statements are equivalent. A is true if and only if B is true. In this case, we would have to show that if statement 1 is true, then statement 2 is true. This would have to be repeated for every pair of distinct statements in the theorem, and would amount to 30 little proofs that together constitute the proof of the theorem. The proof ends up as a cycle of logical inferences.