
theorem
  for G1 being _Graph, E being set, G2 being reverseEdgeDirections of G1, E
  holds G1.loops() = G2.loops()
proof
  let G1 be _Graph, E be set, G2 be reverseEdgeDirections of G1, E;
  now
    let e be object;
    hereby
      assume e in G1.loops();
      then consider v being object such that
        A1: e Joins v,v,G1 by Def2;
      e Joins v,v,G2 by A1, GLIB_007:9;
      hence e in G2.loops() by Def2;
    end;
    assume e in G2.loops();
    then consider v being object such that
      A2: e Joins v,v,G2 by Def2;
    e Joins v,v,G1 by A2, GLIB_007:9;
    hence e in G1.loops() by Def2;
  end;
  hence thesis by TARSKI:2;
end;
