
theorem
  for G being _Graph holds G is loopless iff
    for v being Vertex of G holds not v,v are_adjacent
proof
  let G be _Graph;
  hereby
    assume A1: G is loopless;
    let v be Vertex of G;
    assume v,v are_adjacent;
    then ex e being object st e Joins v,v,G by CHORD:def 3;
    hence contradiction by A1, GLIB_000:18;
  end;
  assume A2: for v being Vertex of G holds not v,v are_adjacent;
  now
    let v be object;
    given e being object such that
      A3: e Joins v,v,G;
    reconsider v0 = v as Vertex of G by A3, GLIB_000:13;
    v0,v0 are_adjacent by A3, CHORD:def 3;
    hence contradiction by A2;
  end;
  hence thesis by GLIB_000:18;
end;
