
theorem
  for G being _Graph holds G is loopfull iff
    for v being Vertex of G holds v,v are_adjacent
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;
    thus v,v are_adjacent by A2, CHORD:def 3;
  end;
  assume A3: for v being Vertex of G holds v,v are_adjacent;
  let v be Vertex of G;
  consider e being object such that
    A4: e Joins v,v,G by A3, CHORD:def 3;
  take e;
  thus thesis by A4;
end;
