
theorem Th1:
  for G being _Graph holds G is loopfull iff
    for v being Vertex of G ex e being object st e DJoins v,v,G
proof
  let G be _Graph;
  hereby
    assume A1: G is loopfull;
    let v be Vertex of G;
    consider e being object such that
      A2: e Joins v,v,G by A1;
    take e;
    thus e DJoins v,v,G by A2, GLIB_000:16;
  end;
  assume A3: for v being Vertex of G ex e being object st e DJoins v,v,G;
  let v be Vertex of G;
  consider e being object such that
    A4: e DJoins v,v,G by A3;
  take e;
  thus e Joins v,v,G by A4, GLIB_000:16;
end;
