
theorem Th47:
  for G being _Graph holds G is loopless iff G.loops() = {}
proof
  let G be _Graph;
  now
    assume G.loops() <> {};
    then consider e being object such that
      A1: e in G.loops() by XBOOLE_0:def 1;
    consider v being object such that
      A2: e Joins v,v,G by A1, Def2;
    thus G is non loopless by A2, GLIB_000:18;
  end;
  hence G is loopless implies G.loops() = {};
  now
    assume G is non loopless;
    then consider v being object such that
      A3: ex e being object st e Joins v,v,G by GLIB_000:18;
    consider e being object such that
      A4: e Joins v,v,G by A3;
    thus G.loops() <> {} by A4, Def2;
  end;
  hence thesis;
end;
