Suppose Vojtech Jarnk had an evil twin, named Stanislaw, who designed a divideand-conquer algorithm for finding minimum

Suppose Vojtech Jarnìk had an evil twin, named Stanislaw, who designed a divideand-conquer algorithm for finding minimum spanning trees. Suppose G is an undirected, connected, weighted graph, and, for the sake of simplicity, let us further suppose that the weights of the edges in G are distinct. Stanislaw’s algorithm, MinTree(G), is as follows: If G is a single vertex, then it just returns, outputting nothing. Otherwise, it divides the set of vertices of G into two sets, V₁ and V₂, of equal size (plus or minus one vertex). Let e be the minimum-weight edge in G that connects V₁ and V₂. Output e as belonging to the minimum spanning tree. Let G₁ be the subgraph of G induced by V₁ (that is, G₁ consists of the vertices in V₁ plus all the edges of G that connect pairs of vertices in V₁). Similarly, let G₂ be the subgraph of G induced by V₂. The algorithm then recursively calls MinTree(G₁) and MinTree(G₂). Stanislaw claims that the edges output by his algorithm are exactly the edges of the minimum spanning tree of G. Prove or disprove Stanislaw’s claim.