reserve G, G1, G2 for _Graph, H for Subgraph of G;

theorem
  G is connected iff the_Vertices_of G.allSpanningTrees() = {the_Vertices_of G}
proof
  hereby
    assume A1: G is connected;
    A2: the_Vertices_of G.allSpanningForests() = { the_Vertices_of G } by Th123
;
    G.allSpanningTrees() = G.allConnectedSG()/\G.allSpanningForests() by Th170;
    then G.allSpanningTrees() c= G.allSpanningForests() by XBOOLE_1:17;
    then the_Vertices_of G.allSpanningTrees() c= { the_Vertices_of G }
      by A2, GLIBPRE1:115;
    hence the_Vertices_of G.allSpanningTrees() = { the_Vertices_of G }
      by A1, ZFMISC_1:33;
  end;
  assume the_Vertices_of G.allSpanningTrees() = { the_Vertices_of G };
  hence thesis;
end;
