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

theorem
  the_Edges_of G = G.loops() iff SubtreeRel(G) = id G.allTrees()
proof
  set H = the removeLoops of G;
  hereby
    assume the_Edges_of G = G.loops();
    then A1: H is edgeless;
    thus SubtreeRel(G) = SubtreeRel(H) by Th164
      .= id H.allTrees() by A1, Th163
      .= id G.allTrees() by Th146;
  end;
  assume A2: SubtreeRel(G) = id G.allTrees();
  SubtreeRel(H) = SubtreeRel(G) by Th164
    .= id H.allTrees() by A2, Th146;
  then H is edgeless by Th163;
  hence thesis by GLIBPRE1:70;
end;
