
theorem Th45:
  for G being _Graph, e being object
  holds e in G.loops() iff ex v being object st e DJoins v,v,G
proof
  let G be _Graph, e be object;
  hereby
    assume e in G.loops();
    then consider v being object such that
      A1: e Joins v,v,G by Def2;
    take v;
    thus e DJoins v,v,G by A1, GLIB_000:16;
  end;
  given v being object such that
    A2: e DJoins v,v,G;
  e Joins v,v,G by A2, GLIB_000:16;
  hence thesis by Def2;
end;
