  We present a strengthened version of a lemma due to Bondy and Lov\'asz. This
lemma establishes the connectivity of a certain graph whose nodes correspond to
the spanning trees of a 2-vertex-connected graph, and implies the k=2 case of
the Gy\H{o}ri-Lov\'asz Theorem on partitioning of k-vertex-connected graphs.
Our strengthened version constructively proves an asymptotically tight O(|V|^2)
bound on the worst-case diameter of this graph of spanning trees.
