
theorem
  for G1 being _Graph, G2 being removeLoops of G1
  for W1 being minlength Walk of G1 holds W1 is Walk of G2
proof
  let G1 be _Graph, G2 be removeLoops of G1;
  let W1 be minlength Walk of G1;
  reconsider P = W1 as Path of G1 by CHORD:36;
  now
    assume P.edges() meets G1.loops();
    then consider v,e being object such that
      A1: e Joins v,v,G1 & P = G1.walkOf(v,e,v) by GLIB_009:57;
    reconsider v as Vertex of G1 by A1, GLIB_000:13;
    A2: P.first() = v & P.last() = v & len P = 3 by A1, GLIB_001:14,15;
    len G1.walkOf(v) = 1 by GLIB_001:13;
    hence contradiction by A2, GLIB_001:13, CHORD:def 2;
  end;
  hence thesis by Th38;
end;
